text:
\begin{document}

\title[The space of compact self-shrinking solutions]{The space of compact self-shrinking solutions to the Lagrangian Mean Curvature Flow in $\mathbb C^2$}
\author{Jingyi Chen and John Man Shun Ma}
\thanks {The first author is partially supported by an NSERC grant.}

\address{Department of Mathematics,
The University of British Columbia,
Canada}
\email{jychen@math.ubc.edu.ca}
\email{johnma@math.ubc.edu.ca}

\begin{abstract}
Let $F_n :(\Sigma, h_n) \to \mathbb C^2$ be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics $\{h_n\}$ converges smoothly to a Riemannian metric $h$. We show that a subsequence of $\{F_n\}$ converges smoothly to a branched conformally immersed Lagrangian self-shrinker $F_\infty : (\Sigma, h)\to \mathbb C^2$. When the area bound is less than $16\pi$, the limit $F_\infty$ is an embedded torus. When the genus of $\Sigma$ is one, we can drop the assumption on convergence $h_n\to h$. When the genus of $\Sigma$ is zero,  we show that there is no branched immersion of $\Sigma$ as a Lagrangian shrinker, generalizing the rigidity result of \cite{Sm0} in dimension two by allowing branch points.
\end{abstract}

\date{June 24, 2014}

\maketitle

\section{Introduction}
Let $\Sigma$ be an orientable smooth $m$ dimensional manifold without boundary and let $F$ be an immersion of $\Sigma$ into $\mathbb R^{m+k}$. The smooth mapping $F$ is called a {\sl self-shrinker} if it satisfies the equation
\begin{equation} \label{self similar eqn}
\vec H = -F^\perp,
\end{equation}
where $\vec H$ is the mean curvature vector and $F^\perp$ is the normal component of the position vector $F$. A self-shrinker gives rise to a solution, via the scaling  $F_t = \sqrt{1-2 t}\, F$ for $0 \leq t <1/2$, of the mean curvature flow (MCF) in ${\mathbb R}^{m+k}$:
\[\bigg(\frac{\partial F_t}{\partial t}\bigg)^\perp = \vec H(F_t).\]
MCF typically develops singularities in finite time. By parabolic rescaling at a singularity, Huisken's monotonicity formula \cite{H} shows that a subsequence of the mean curvature flow converges to a self-shrinker when the singularity is of type I. Ultimately, the classification of self-shrinkers would be important in understanding the MCF.

When $m+k=2m$, we can identify $\mathbb R^{2m}$ with $\mathbb C^m$ and consider the Lagrangian immersions. It is shown in \cite{Sm1} that the Lagrangian condition is preserved along the MCF. Therefore, the shrinkers arise from Lagrangian MCF are Lagrangian submanifolds. When $m=1$, the Lagrangian condition is automatically satisfied by smooth curves and self shrinking solutions are studied in \cite{AL}.

In this paper, we will restrict our attention to compact Lagrangian self shrinkers in ${\mathbb C}^2$. For arbitrary $m$, Smoczyk \cite{Sm0} showed that there is no Lagrangian self-shrinking immersion in $\mathbb C^m$ with first Betti number equals zero. In particular,  there is no immersed Lagrangian self shrinking sphere in $\mathbb C^2$. To establish compactness properties of the moduli space of compact Lagrangian shrinkers, it is crucial to generalize Smoczyk's result to {\it branched} Lagrangian immersions as the limit of a sequence of immersions may not be an immersion anymore.

Indeed, the rigidity holds for branched Lagrangian shrinking spheres in ${\mathbb C}^2$, we have:

\begin{thm} \label{no S2 lag shrinker}
There does not exist any branched conformal Lagrangian self-shrinking sphere in $\mathbb C^2$.
\end{thm}

At any immersed point,  the mean curvature form $\alpha_H = \iota_{\vec H} \omega$ of the Lagrangian shrinker satisfies a pair of differential equations (\ref{da=0}) and  (\ref{d*a eqn}) that form a first order elliptic system. The key ingredient in the proof of Theorem \ref{no S2 lag shrinker} is to show that $\alpha_H$ extends smoothly across the branch points. The self-shrinker equation (\ref{self similar eqn}) yields a $L^\infty$ bound on  $\alpha_H$ and this is useful to show that $\alpha_H$ satisfies the first order elliptic system as a weak solution distributionally. Smoothness of the extended mean curvature form then follows from the elliptic theory.

The main application of Theorem \ref{no S2 lag shrinker} is to derive compactness results of compact Lagrangian self-shrinkers.

Let $\Sigma$ be a fixed compact oriented smooth surface and $F_n:\Sigma \to \mathbb C^2$ be a sequence of Lagrangian immersed self-shrinkers in $\mathbb C^2$.  Let $h_n$ be the Riemannian metric on $\Sigma$ which is conformal to the pull back metric $F_n^*\il \cdot, \cdot\li$  on $\Sigma$ such that either it has constant Gauss curvature $-1$ if the genus of $\Sigma$ is greater than one, or $(\Sigma , h_n)$ is $ \mathbb C / \{1, a+ bi\}$ with the flat metric, where $-\frac{1}{2} < a\leq \frac{1}{2}$, $b\geq 0$, $a^2 + b^2 \geq 1$ and $a\geq 0$ whenever $a^2 + b^2  =1$. It is well known that the Moduli space of the conformal structures on $\Sigma$ are parametrized by metrics of the above form.

We now state our compactness result.

\begin{thm} \label{main theorem}
Let $F_n :(\Sigma, h_n) \to \mathbb C^2$ be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound $\Lambda$. Suppose that the sequence of metrics $\{h_n\}$ converges smoothly to a Riemannian metric $h$ on $\Sigma$. Then there is a subsequence of $\{F_n\}$ that converges smoothly to a branched conformally immersed Lagrangian self-shrinkers $F_\infty :(\Sigma, h) \to \mathbb C^2$.
\end{thm}

Note that  there is a universal positive lower bound on the extrinsic diameters for two dimensional compact shrinkers (cf. section 4). The limit $F_\infty$ cannot be a constant map.

The proof of Theorem \ref{main theorem} uses the observation that self-shrinkers are in fact minimal immersions into $(\mathbb C^2, G)$, where $G$ is a metric on $\mathbb C^2$ conformal to the Euclidean metric.
The advantage of this viewpoint is that we are then able to use the bubble tree convergence of harmonic maps developed in \cite{P}. In particular, Theorem \ref{no S2 lag shrinker} shows that no bubble is formed during the process and thus the convergence is smooth.

It is interesting to compare Theorem \ref{main theorem} with the compactness results of Colding-Minicozzi \cite{CM1} on self-shrinkers in $\mathbb R^3$ and of Choi-Schoen \cite{CS}, Fraser-Li \cite{FL} on embedded minimal surfaces in three dimensional manifold $N$ (with or without boundary) with nonnegative Ricci curvature. In their cases, they use the local singular compactness theorem (see for example Proposition 2.1 of \cite{CM1}), which says that for any sequence of embedded minimal surfaces $\Sigma_n$ in $N$ with a uniform (global or local) area or genus upper bound, there is an embedded minimal surface $\Sigma$ such that a subsequence of $\Sigma_n$ converges smoothly and locally (with finite multiplicities) to $\Sigma$ away from finitely many points in $\Sigma$. A removable singularity theorem, which is based on the maximum principle
and is true only in the codimension one case, is needed to prove the local singular compactness theorem. Hence similar statement is not available in higher codimensions. For embedded minimal surfaces in a 3-manifold, Colding and Minicozzi have proven deep compactness results (\cite{CM2} and the reference therein).

In the case of arbitrary codimension, the regularity of the limit of a sequence of minimal surfaces (as a stationary varifold  given by Geometric Measure Theory) is a subtle issue. Not much is known except for some special cases, such as Gromov's compactness theorem on pseudo holomorphic curves \cite{G}.

In our situation, we do not assume embeddedness. In fact, if we assume that  $F_n$ is a Lagrangian embedding, then by a result of Whitney (\cite{W}, see also \cite{ALP}), $\Sigma$ must be a torus as the self-intersection number $\Sigma\cdot\Sigma$ is twice of the genus of $\Sigma$ minus 2. When we treat the immersions $F_n$ as conformal harmonic mappings, even impose the assumption that the conformal structures induced from the immersions  stay in a compact domain of the Teichm\"uller space,  the sequence $F_n$ would subsequentially converge only to a {\it branched} immersion on each component in the bubble tree convergence. True branch points might exist for the limiting map $F_\infty$. On the other hand, if $\Lambda<16\pi$ in Theorem \ref{main theorem} then the Willmore energy of each immersion $F_n$ is less than $8\pi$ for shrinkers (cf. \cite{CL}), hence by an inequality of Li-Yau (Theorem 6 in \cite{LY}), $F_n$ must be an embedding and in turn $\Sigma$ is a torus since it is Lagrangian.

When $\Sigma$ has genus one, we can drop the assumption on the convergence of conformal structures in Theorem \ref{main theorem}. Moreover, if the area bound is strictly less than $16\pi$, the limiting surface has no branched points and we have smooth compactness.

\begin{thm} \label{main thm 2}
Let $F_n : T \to \mathbb C^2$ be a sequence of Lagrangian self-shrinking tori with a uniform area upper bound $\Lambda$. Then (i) there is a subsequence of $\{F_n\}$ that converges smoothly to a branched Lagrangian self-shrinking torus $F_\infty:T\to{\mathbb C}^2$, and (ii) when $\Lambda<16\pi$, $F_\infty$ is an embedding.
\end{thm}

Our strategy of proving Theorem \ref{main thm 2} is to rule out the possibility of degeneration of the conformal structures induced by $F_n$ on $T$. To do this we need the general bubble tree convergence results in \cite{CLi}, \cite{CT} that allow the conformal structures degenerate. The key observation is that when the conformal structures degenerate, some homotopically nontrivial closed curves in the torus are pinched to points. Thus the limiting surface is a finite union of spheres (arising from collapsion of the closed curves and from the bubbles at the energy concentration points) which are branched Lagrangian shrinkers, but Theorem \ref{no S2 lag shrinker} forbids this. The desired result then follows from Theorem \ref{main theorem}.

\section{Preliminaries}
Given an immersion $F : \Sigma \to \mathbb R^{m+k}$, let $g$ be the induced metric $F^*\il \cdot, \cdot\li$ on $\Sigma$. In local coordinates $(x^1. \cdots x^m)$ of $\Sigma$, the second fundamental form $A$ and the mean curvature vector $\vec H$ are given by

$$
A_{ij}= \big(\partial_{ij} F \big)^\perp,\ \ \vec H = \sum_{i,j=1}^m g^{ij}A_{ij} = \Delta_g F
$$
where $\Delta_g$ is the Laplace-Beltrami operator in the induced metric $g$.

Let $G$ be a metric on $\mathbb R^{m+k}$ defined by
\begin{equation} \label{dfn of G}
G(\cdot, \cdot) = e^{-\frac{|x|^2}{m}} \il \cdot , \cdot\li  .
\end{equation}

The following lemma is well-known and is proved by Angenent \cite{A} in the case of hypersurface. The proof can be generalized to arbitrary codimension.

\begin{lem} \label{self similar iff minimal wrt g}
The immersion $F$ satisfies equation (\ref{self similar eqn}) if and only if $F$ is a minimal immersion with respect to the metric $G$ (\ref{dfn of G}) on $\mathbb R^{m+k}$.
\end{lem}

Next we consider immersions $F:\Sigma \to \mathbb C^m$. We identify $\mathbb R^{2m}$ with $\mathbb  C^m$ by $z^i = u^i + \sqrt{-1} v^i, i=1,\cdots,m$. The map $F$ is called a Lagrangian immersion if $F^*\omega=0$, where
$$
\omega = \sum_{i=1}^m du^i \wedge dv^i
$$
is the standard symplectic form on $\mathbb C^m$. Let $J$ be the standard complex structure on $\mathbb C^m$. It is known that $\omega$, $J$ and $\il \cdot, \cdot \li$ on $\mathbb C^m$ are related by
\begin{equation} \label{compatibility of g, w, J}
\il JX, JY\li = \il X, Y\li,\ \ \omega(X, Y) = \il JX, Y\li
\end{equation}
for any $X,Y\in T_x{\mathbb C}^m, x\in {\mathbb C}^m$. Thus $F$ is Lagrangian if and only if $J$ sends the tangent vectors of $\Sigma$ to the normal vectors in ${\mathbb C}^m$.

In particular, by equation (\ref{compatibility of g, w, J}), $J\vec H(x)$ is tangent to $F(\Sigma)$ at $x$ for any Lagrangian immersion $F$. The mean curvature form $\alpha_H$ is the 1-form on $\Sigma$ defined by:  for all $x\in \Sigma$ and $Y\in T_x \Sigma$,
\begin{equation} \label{a dfn}
\alpha_H (Y)= \omega (\vec H (x), (F_*)_xY)= g (J\vec H (x), (F_*)_xY ).
\end{equation}
Let $d$ denotes the exterior differentiation of $\Sigma$. It is shown in \cite{D} that $\alpha_H$ is closed:
\begin{equation} \label{da=0}
d\alpha_H = 0\ .
\end{equation}

If $F$ is also a self-shrinker, we have the following equation for $\alpha_H$ (\cite{CL}, p.1521). For the convenience of the reader, we include a proof.

\begin{lem}  \label{d^*a lemma} Let $F$ be a Lagrangian self-shrinker. Then $\alpha_H$ satisfies
\begin{equation} \label{d*a eqn}
d^* \alpha_H =  -\frac 12 \alpha_H (\nabla_g |F|^2) ,
\end{equation}
where $d^*$ is the formal adjoint of $d$ on $\Sigma$ with respect to $g$.
\end{lem}

\begin{proof}
In local coordinates,
$$
\alpha_H = \sum_{i=1}^m(\alpha_H)_i dx^i
$$
where the coefficients are given by
\begin{eqnarray*}
(\alpha_H)_i &=&  \il J\vec H, \partial_i F\li \\
&=& -\il JF^\perp , \partial _i F\li  \\
&=& -\il JF, \partial _i F\li.
\end{eqnarray*}
Note that we have used the self-shrinker equation (\ref{self similar eqn}) in the second equality. Now fix a point $p\in \Sigma$ and take the normal coordinates at $p$. At $p$, $g_{ij} = g^{ij} = \delta_{ij}$ and $\Gamma_{ij}^k = 0$. We now calculate at $p$,
\[\begin{split}
d^*\alpha_H &= -\sum_{i,j=1}^m g^{ij} (\alpha_H)_{i;j} \\
&= - \sum_{i=1}^m (\alpha_H)_{i,i} \\
&=  \sum_{i=1}^m \partial_i \il JF, \partial _i F \li \\
&=  \sum_{i=1}^m \left(\il J \partial_i F, \partial _i F \li  + \il JF , \partial^2_{ii} F \li  \right)
\end{split}\]
Using $J \partial_i F \perp \partial _i F$ and $\partial^2_{ii} F = A_{ii}$ for each $i$ at $p$,
\[
\begin{split}
d^*\alpha_H &=  \sum_{i=1}^m \il JF, A_{ii} \li \\
&= \il JF,\vec H \li \\
&= -\il F, J\vec H\li \\
&= - \alpha_H (F^\top) \\
&= -\frac 12 \alpha_H(\nabla_g |F|^2),
\end{split}
\]
where $F^\top$ is the tangential component of $F$ and we have used
$$
F^\top=\sum^m_{i=1}\langle F, \partial_i F\rangle \,\partial_i F = \frac{1}{2}\nabla_g|F|^2
$$
at $p$.
\end{proof}

\section{ Lagrangian conformal branched immersion}
From now on we consider $m=2$. Let $(\Sigma, g_0)$ be a smooth Riemann surface. A smooth map $F :\Sigma \to \mathbb C^2$ is called a branched conformal immersion if
\begin{enumerate}
\item there is a discrete set $B \subset \Sigma$ such that
\[F: \Sigma \setminus B \to \mathbb C^2\]
 is an immersion,
\item there is a function $\lambda : \Sigma \to [0,\infty)$ such that $g:= F^* \il \ , \ \li = \lambda  g_0$ on $\Sigma$, where $\il \ , \ \li$ is the Euclidean metric on $\mathbb C^2$ and $\lambda$ is zero precisely at $B$, and
\item the second fundamental form $A$ on $\Sigma\setminus B$ satisfies $|A|_g \in L^2 (K\setminus B, d\mu)$ for any compact domain $K$ in $\Sigma$, where $|\cdot |_g$ and $d\mu$ are respectively the norm and the area element with respect to $g$ (Note $g$ defines a Riemannian metric on $\Sigma\setminus B$ but not on $\Sigma$).
\end{enumerate}

Elements in $B$ are called the branch points of $F$. A conformal branched immersion $F:\Sigma \to \mathbb C^2$ is called Lagrangian (resp. self-shrinking) if it is Lagrangian (resp. self-shrinking) when restricted to $\Sigma\setminus B$. Note that when $F$ is Lagrangian, the mean curvature form $\alpha_H$ is defined only on $\Sigma \setminus B$.

The following proposition is the key result on removable singularity of $\alpha_H$. Note that in this proposition we do not assume $\Sigma$ to be closed.

\begin{prop} \label{a extend across B}
Let $F:\Sigma \to \mathbb C^2$ be a branched conformal Lagrangian self-shrinker with the set of branch points $B$. Then there is a smooth one form $\tilde \alpha$ on $\Sigma$ which extends $\alpha_H$ and $d\tilde \alpha=0$ on $\Sigma$.
\end{prop}

\begin{proof}
The result is local, so it suffices to consider $\Sigma$ to be the unit disc $\mathbb D$ with a branch point at the origin only. Let $(x, y)$ be the local coordinate of $\mathbb D$. We write $\alpha_H  = a dx + bdy$ for some smooth function $a$ and $b$ on the punctured disk $\mathbb D^* = \{ z\in \mathbb D: z\neq 0\}$. Let
\begin{equation} \label{dfn of delta and D}
\text{div}( \alpha_H) =   \frac{\partial a}{\partial x} + \frac{\partial b}{\partial y}   , \ \ \ \ D  = \bigg(\frac{\partial }{\partial x}, \frac{\partial }{\partial y} \bigg)
\end{equation}
be the divergence and the gradient with respect to $\delta_{ij}$. As $F$ is conformal, $g_{ij} = \lambda \delta_{ij}$, where $\lambda = \frac{1}{2}| DF|^2$ and $\delta_{ij}$ is the euclidean metric on $\mathbb D$. By restricting to a smaller disk if necessary, we assume that the image $|F|$ and $\lambda$ are bounded.  By equation (\ref{self similar eqn}) we have $$|\vec H| \leq |F|,$$ where $|\cdot|$ is taken with respect to $\il \cdot, \cdot\li$ on $\mathbb C^2$. As $g = F^*\il \cdot, \cdot\li$, using equation (\ref{a dfn}) we see that
$$
|\alpha_H|_g \leq C\,\,\text{ on }\mathbb D^*.
$$
 Using $g_{ij} = \lambda \delta_{ij}$ and $n=2$, we have
$$
\nabla_g = \lambda^{-1} D\, \, \text{and}\, \,d^*_g  = - \lambda^{-1} \text{div}.
$$
Hence equation (\ref{d*a eqn}) is equivalent to
\begin{equation} \label{d*a eqn euclidean}
\text{div}( \alpha_H) =  \frac 12 \alpha_H (D |F|^2 ).
\end{equation}
Moreover, as $|\alpha_H|_g = \sqrt{\lambda^{-1} (a^2 + b^2)}$, we also have
$$|a|, |b| \leq  |\alpha_H|_g \sqrt\lambda\leq C\sqrt\lambda.$$
Thus $|a|, |b|$ are bounded on $\mathbb D^*$. To simplify notations, let
$$P = \frac 12 \alpha_H (D|F|^2).$$
Note that $P$ is also bounded on $\mathbb D ^*$.

Both equations (\ref{da=0}) and (\ref{d*a eqn euclidean}) are satisfied pointwisely in $\mathbb D^*$. We now show that they are satisfied in the sense of distribution on $\mathbb D$. That is, for all test functions $\phi \in C^\infty_0(\mathbb D)$,
\begin{equation} \label{da=0 dist}
\int_\mathbb{D} \alpha_H \wedge d\phi = 0
\end{equation}
and
\begin{equation} \label{d* a eqn dist}
\int_\mathbb{D} \alpha_H (D\phi) dxdy =  - \int_\mathbb{D}P \phi dxdy .
\end{equation}
Note that all the integrands in equation (\ref{da=0 dist}) and (\ref{d* a eqn dist}) are integrable, since $a, b$ and $P$ are in $L^\infty (\mathbb D)$.

First we show (\ref{da=0 dist}). Let $\psi_r \in C^\infty_0(\mathbb D)$, $r<1/2$, be a cutoff function such that $0\leq \psi \leq 1$,  $|D\psi| \leq 2/r$ and
\[\psi(x) = \begin{cases} 1 & \text{when }|x|\geq 2r,\\
 0 & \text{when } |x|\leq r.
\end{cases}\]
Then $\phi \psi_r \in C^\infty_0(\mathbb D^*)\cap C^\infty(\mathbb D)$. Using Stokes' theorem and equation (\ref{da=0}), we have
$$
 0 =\int_{\mathbb D ^*} d (\phi\psi_r \alpha_H) =  \int_{\mathbb D^*} d(\phi \psi_r)\wedge   \alpha_H  .
$$
This implies
\begin{equation} \label{da =0 dist r}
 \int_{\mathbb D^* } \psi_r  \alpha_H \wedge d\phi= -\int_{\mathbb D^*} \phi  \alpha_H \wedge  d\psi_r .
\end{equation}
Since $\psi_r \to 1$ on $\mathbb D^* $ as $r\to 0$ and $\alpha_H$, $d\phi$ are bounded,
$$
\lim_{r\to 0} \int_{\mathbb D^*} \psi_r \alpha_H \wedge d\phi = \int_{\mathbb D^*} \alpha_H \wedge d\phi =\int_\mathbb{D} \alpha_H \wedge d\phi
$$
by Lebesque's dominate convergence theorem. To estimate the right hand side of equation (\ref{da =0 dist r}), note that as $d\psi_r$ has support on $\mathbb D_{2r} \setminus \mathbb D_r$, where $\mathbb D_s$ denotes the disk of radius $s$. Hence
\begin{equation} \label{estimate of da = 0}
\bigg| \int_\mathbb{D} \phi \alpha_H \wedge d\psi_r \bigg|\leq \frac{4 C \sup |\phi|}{r} \int_{\mathbb D_{2r}\setminus \mathbb D_r} dx  = 12\pi C \sup|\phi| r \to 0
\end{equation}
as $r \to 0$. Thus equation (\ref{da=0 dist}) holds.

To show equation (\ref{d* a eqn dist}), we use the same cutoff function $\psi_r$. Then $\phi \psi_r \in C^\infty_0(\mathbb D^*)\cap C^\infty(\mathbb D)$. By the divergence theorem,
$$
0 = \int_{\mathbb D^*} \text{div}(\phi\psi_r \alpha_H) \,dxdy
= \int_{\mathbb D^*} \alpha_H \big( D(\phi\psi_r ) \big) \, dxdy + \int_{\mathbb D^*} \phi\psi_r \text{div}(\alpha_H) \,dxdy .
$$
Now we use equation (\ref{d*a eqn euclidean}) to conclude
\begin{equation} \label{d*a eqn dist proof}
-\int_{\mathbb D^*} P\phi\psi_r \,dxdy = \int_{\mathbb D^*} \psi_r \alpha_H (D\phi) \,dxdy + \int_{\mathbb D^*} \phi \alpha_H (D\psi_r)\,dxdy .
\end{equation}
Similarly, we can estimate the second term on the right hand side of equation (\ref{d*a eqn dist proof}) as for equation (\ref{estimate of da = 0}):
\begin{equation}\label{estimate of d^*a}
\bigg| \int_{\mathbb D^*} \phi \alpha_H (D\psi_r)\,dxdy\bigg| \leq  12\pi C \sup|\phi| r .
\end{equation}
Using Lebesque's dominate convergence theorem again, we can set $r\to 0$ in equation (\ref{d*a eqn dist proof}) to arrive at equation (\ref{d* a eqn dist}).

Writing $\alpha_H = a dx + bdy$, equations (\ref{da=0 dist}) and (\ref{d* a eqn dist}) are equivalent to
\begin{equation} \label{first dist eqn}
\int_\mathbb{D} \bigg( a \frac{\partial \phi}{\partial y} - b\frac{\partial \phi}{\partial x} \bigg) dxdy = 0 ,
\end{equation}
\begin{equation} \label{second dist eqn}
\int_\mathbb{D} \bigg( a \frac{\partial \phi}{\partial x} + b\frac{\partial \phi}{\partial y} \bigg) dxdy = -\int_\mathbb{D}P \phi \, dxdy ,
\end{equation}
for any test functions $\phi \in C_0^\infty(\mathbb D)$.

For any $\psi \in C^\infty_0(\mathbb D)$, set $\phi = \frac{\partial \psi}{\partial y}$ in equation (\ref{first dist eqn}), $\phi = \frac{\partial \psi}{\partial x}$ in equation (\ref{second dist eqn}) and cancel the cross term $b\frac{\partial ^2 \psi}{\partial x\partial y}$ by taking summation of the two, we have
\[
\int_\mathbb{D} a \,\Delta \psi \,dxdy=  -\int_\mathbb{D} P\, \frac{\partial \psi}{\partial x} \,dxdy
\]
where $\Delta$ is the Laplace operator in the euclidean metric on ${\mathbb D}$.

Similarly, set $\phi = \frac{\partial \psi}{\partial x}$ in equation (\ref{first dist eqn}), $\phi = \frac{\partial \psi}{\partial y}$ in equation (\ref{second dist eqn}) and take the difference of the two equations, we obtain
\[
\int_\mathbb{D} b \,\Delta \psi \,dxdy=  -\int_\mathbb{D} P\, \frac{\partial \psi}{\partial y} \,dxdy .
\]

We conclude now that $a$ and $b$ satisfy
\begin{equation} \label{a, b dist eqn}
\begin{split}
\Delta \,a &= \frac{\partial P}{\partial x} , \\
\Delta \,b &= \frac{\partial P}{\partial y}
\end{split}
\end{equation}
on $\mathbb D$ in the sense of distribution. Now we apply the elliptic regularity theory for distributional solutions. As $F$ is smooth and $\alpha\in L^\infty (\mathbb D)$, we have $P \in L^2(\mathbb D)$. Hence the right hand side of equation (\ref{a, b dist eqn}) is in $H^{loc}_{-1}(\mathbb D)$. By the local regularity theorem (\cite{F}, Theorem 6.30), we have $a, b\in H_1^{loc} (\mathbb D)$. This implies $P \in H_1^{loc} (\mathbb D)$. Using this, we see that the right hand side of equation (\ref{a, b dist eqn}) is in $H_0^{loc}(\mathbb D)$. By the same local regularity theorem again, these implies $a, b\in H_2^{loc }(\mathbb D)$. Thus we can iterate this argument and see that $a, b \in H_s^{loc}(\mathbb D)$ for all integers $s$. By the Sobolev embedding theorem we have $a, b \in C^\infty (\mathbb D)$. Hence $\alpha_H$ can be extended to a smooth one form $\tilde \alpha$ on $\mathbb D$ and $d\tilde \alpha=0$ is satisfied on $\mathbb D$.
\end{proof}

Now we proceed to the proof of Theorem \ref{no S2 lag shrinker}.

\begin{proof}
 Let $F :\mathbb S^2 \to \mathbb C^2$ be a branched conformal Lagrangian self-shrinker with branch points $b_1, \cdots, b_k$. By Proposition \ref{a extend across B}, there is a smooth 1-form $\tilde \alpha$ on $\mathbb S^2$ such that $\tilde \alpha = \alpha_H$ on $\mathbb S^2\setminus B$. As $\tilde \alpha$ is closed and the first cohomology group of $\mathbb S^2$ is trivial, there is a smooth function $f$ on $\mathbb S^2$ such that $df = \tilde \alpha$. By equation (\ref{d^*a lemma}), $f$ satisfies
\begin{equation} \label{eqn of f}
 \Delta_g f = \frac 12 \,df (\nabla_g |F|^2)
\end{equation}
on $\mathbb S^2\setminus B$. Note that this equation is elliptic but not uniformly elliptic on $\mathbb S^2\setminus B$. By the strong maximum principle, the maximum of $f$ cannot be attained in $\mathbb S^2\setminus B$ unless $f$ is constant. Let $b\in B$ be a point where $f$ attains its maximum. Let $\mathbb D$ be a local chart around $b$ such that $g_{ij} = \lambda \delta_{ij}$ on $\mathbb D ^*$. As $\Delta_g = \lambda^{-1}\Delta$ and $\nabla_g = \lambda^{-1} D$, equation (\ref{eqn of f}) can be written
\begin{equation} \label{eqn of f eu}
\Delta f =  \frac 12 \,df(D|F|^2 )\ \ \ \  \text{on }\mathbb D^* .
\end{equation}
As $f$ is smooth on $\mathbb D$, equation (\ref{eqn of f eu}) is in fact satisfied on $\mathbb D$. By the strong maximum principle, $f$ is constant as $f$ has an interior maximum at $b$. Hence $\alpha_H = 0$ and $\vec H=0$. This implies that $F$ is a branched minimal immersion in $\mathbb C^2$, which is not possible as $\mathbb S^2$ is compact.
\end{proof}

\section{Proof of Theorem \ref{main theorem}}
Let $F_n$ be a sequence of Lagrangian self-shrinkers which satisfies the hypothesis in the theorem. We need the following two lemmas for proving convergence of  $F_n$.

The first lemma asserts that the images of $F_n$, for all $n$,  stay in a bounded region whose size depends only on the area bound $\Lambda$. In the following we use $B_r$ to denote the open ball in $\mathbb C^2$ with radius $r$ centered at the origin.

\begin{lem} \label{F in U} Let $F_n$ be a sequence of two dimensional self-shrinkers in ${\mathbb R}^4$ with a uniform area upper bound.
Then the image of $F_n$ lies in $B_{R_0}$, where $R_0$ depends only on the area upper bound.
\end{lem}

\begin{proof}
It is proved in \cite{CL} that  for any $m$ dimensional self-shrinker $F :\Sigma^m \to \mathbb R^{m+k}$.
\begin{equation} \label{Delta |F|^2}
\Delta_g |F|^2 =  - 2|F^\perp|^2 + 2m .
\end{equation}
 When $\Sigma$ is compact, there is a point $p\in \Sigma$ where $|F|^2$ attains its minimum. Then we have
$$
\nabla_g |F|^2 = 2 F^\top = 0\ \  \text{  and  }\ \ \Delta_g F \geq 0 \ \ \text{  at  }p.
$$
This implies
$$
0 \leq -2|F|^2 +2m \ \ \text{at $p$}.
$$
Hence $|F|^2 \leq m$ at $p$. Thus each $F_n$ must intersect the ball with radius $\sqrt 2$.

On the other hand, we integrate equation (\ref{F in U}) and use equation (\ref{self similar eqn}) to get
(also derived in \cite{CL})
\begin{equation} \label{W = A}
\mathcal W (F_n) :=\frac{1}{4} \int_\Sigma |H_n|^2 d\mu_n = \frac{1}{2} \mu (F_n) ,
\end{equation}
where $\mu(F_n)$ is the area of the immersion $F_n : \Sigma \to \mathbb R^{m+k}$.

Next we use Simon's diameter estimate (\cite{S}, see also (2.1) in \cite{KLi}), which says that there is a constant $C$ such that
\begin{equation} \label{bound in diam}
  \bigg(\frac{\mu (F_n)}{ \mathcal W(F_n)}\bigg)^{\frac{1}{2}} \leq  \text{diam}F_n(\Sigma) \leq C\big(  \mu (F_n) \mathcal W(F_n)\big)^{\frac{1}{2}} .
\end{equation}
Here diam$F_n(\Sigma)$ is the extrinsic diameter given by
\[
\text{diam}F_n(\Sigma) := \sup_{x, y\in \Sigma} |F_n(x) - F_n(y)|.
\]
As the areas are uniformly bounded, the images of $F_n$ all lie in $B_{R_0}$ for some $R_0$ depending only on the area upper bound.
\end{proof}

Let $U = B_{R_0+1}$ endowed with the metric $G$ given by
\begin{equation} \label{conformal metric on C^2}
G = e^{-\frac{|x|^2}{2}} \il \cdot, \cdot\li  .
\end{equation}

The next lemma enables us to apply the results in \cite{P}  for harmonic maps into a compact Riemannian manifold.

\begin{lem} \label{U embed in N}
There is a compact Riemannian manifold $(N, \overline g)$ such that $(U, G)$ isometrically embedded into $(N, \overline g)$.
\end{lem}

\begin{proof}
Let $d = \frac{1}{R_0+1}$ and $N$ is the disjoint union of $B_{R_0 + 2}$ and $B_d$, with the identification that $x\sim y$ if and only if $y = \frac{x}{|x|^2}$ by the inversion. The manifold $N$ is compact, as it can be identified as the one point compactification of $\mathbb C^2$ via the stereographic projection.  Let $g_1$ be any metric on $B_d$. Let $\rho_1, \rho_2 \in C^\infty(N)$ be a partition of unity subordinate to the open cover $\{ B_{R_0+2}, B_d \}$ in $N$ and define a Riemannian metric on $N$ by $$\overline g = \rho_1 G + \rho_2 g_1.$$ As
$$B_{R_0+2} \cap B_d = \{x \in B_{R_0+2}:  R_0+1 < |x| < R_0+2\},$$
$\bar g = \rho_1 G + \rho_2 g_1 = G$ on $B_{R_0+1}\subset B_{R_0+2}$. Thus the inclusion $U\subset B_{R_0+1} \subset N$ is an isometric embedding of $U$.
\end{proof}

Now we are ready to prove Theorem \ref{main theorem}.

\begin{proof} As $F_n(\Sigma) \subset U$ for all $n$, we also treat $F_n$ as a map with image in $N$. As $F_n$ is a conformal minimal immersion with respect to the metric $h_n$ on $\Sigma$ and $G = \bar g$ on $U$, $F_n :(\Sigma , h_n) \to (N, \bar g)$ is a sequence of harmonic maps. The area of $F_n(\Sigma)$ in $(N,\bar{g})$ is
$$
\tilde{\mu}(F_n) = \int_\Sigma e^{-\frac{|F_n|^2}{2}}d\mu_{F_n^*\langle\cdot,\cdot\rangle}.
$$
Therefore, as $e^{-\frac{|F_n|^2}{2}} \leq 1$ we have
$$
\tilde \mu (F_n) \leq \mu (F_n)<\Lambda.
$$
As $F_n:(\Sigma,h_n)\to (N,\bar g)$ is conformal, the area $\tilde \mu (F_n)$ in $N$ is the same as the Dirichlet energy:
\begin{equation*}
E_{h_n,\bar g}(F_n)=\tilde \mu (F_n) <\Lambda.
\end{equation*}
By assumption, the sequence of metrics $h_n$ converges to $h$. Hence we can apply the theory of bubble tree convergence of harmonic maps developed in \cite{P}. In particular, by passing to subsequence if necessary, there is a finite (possibly empty) set $S\subset \Sigma$  and a harmonic map $F_\infty :(\Sigma, h) \to (N, \bar g)$ such that $F_n \to F_\infty$ locally in $C^\infty(\Sigma \setminus S)$ \footnote{In \cite{SU1}, it is shown that the convergence is in locally $C^1(\Sigma\setminus S)$. As each $F_n : (\Sigma , h_n) \to (\mathbb C^2, G)$ is harmonic and $h_n \to h$, by the standard elliptic estimates (Chapter 6 of \cite{GT}) and a bootstrapping argument, there are constants $C(m)$ such that
$$
||F_n||_{C^m} \leq C(m)
$$
for all $n \in \mathbb N$ . Using Arzela-Ascoli Theorem and picking a diagonal subsequence, one shows that a subsequence of $\{F_n\}$ converges locally smoothly to a smooth mapping $\Sigma \to \mathbb C^2$, which is $F_\infty$. }. For each $x \in S$, there is a finite sequence of nontrivial harmonic maps $s_{x, i} :\mathbb S^2 \to N$, $i=1, \cdots, n_x$ such that the renormalized maps $F_{n, I}$ (See \cite{P} for the details) of $F_n$ converges to the harmonic bubble tree map formed by $F_\infty$ and $s_{x,i}$. In particular, we have the Hausdorff convergence
$$
F_n(\Sigma) \to F_\infty(\Sigma)\cup \bigcup_{x\in S}  \bigcup_{i} s_{x, i} (\mathbb S^2).
$$
As a result, $F_\infty$ and $s_{x,i}$ all have image in $\overline U \subset N$.

The harmonic maps $s_{x,i}: \mathbb S^2 \to \overline U$ are branched conformal minimal immersions. From the construction of bubble tree convergence \cite{P},  a bubble at $x\in \Sigma$ is constructed by considering the renormalization
$$
\tilde f_n(z) := f_n(\exp_x( \lambda_n z + c_n)),
$$
where $c_n, \lambda_n \to 0$ as $n\to\infty$. As the normalization is only on the domain, $\tilde f_n$ are also Lagrangian. The regularity results in \cite{SU} show that there is a finite set $\tilde S \subset \mathbb C$ such that $\tilde f_n$ converges locally in $C^\infty(\mathbb C \setminus \tilde S)$ to a harmonic map $\tilde f_\infty : \mathbb S^2 \to N$. Thus the bubble $\tilde f_\infty$ is Lagrangian. This applies to each level of the bubble tree, so all bubbles formed are also Lagrangian. Using Lemma \ref{U embed in N} and Lemma \ref{self similar iff minimal wrt g}, each $s_{x, i}$ is a branched Lagrangian self-shrinker in $\mathbb C^2$. By Theorem \ref{no S2 lag shrinker}, such an immersion does not exist. Thus there is no bubble formed in the convergence process (that is, $S$ is empty), and the convergence $F_n \to F_\infty$ is in $C^\infty(\Sigma)$.

By Theorem 3.3 in \cite{SU}, as each $F_n : \Sigma \to N$ is nontrivial by definition, there is an $\epsilon >0$ such that $E_{h_n,\bar g}(F_n) \geq \epsilon$ for all $n\in \mathbb N$. Using the energy identity \cite{J}, we have
\begin{equation}
 E_{h_n,\bar g}(F_n) \to E_{h,\bar g}(F_\infty) \ \ \ \text{as }n\to \infty .
\end{equation}
Hence $E_{h,\bar g}(F_\infty) \geq \epsilon$ and $F_\infty$ is nontrivial (One can also use the estimate of diameter and the Hausdorff convergence as in the proof of Theorem \ref{main thm 2} in the next section to show that $F_\infty$ is nontrivial).

As the convergence $F_n \to F_\infty$ is in $C^\infty(\Sigma)$ and the metrics $h_n$ converges to $ h$ smoothly, the harmonic map $F_\infty : (\Sigma, h) \to (N, \bar g)$ is also conformal. Thus, $F_\infty$ is a branched conformal minimal immersion from $(\Sigma,h)$ into $(N,\bar g)$. It is also a Lagrangian immersion. Since the image of $F_\infty$ lies in $\overline U$, we also view $F_\infty$ as a branched Lagrangian minimal immersion in $\mathbb C^2$ with respect to the metric $G$ defined by (\ref{conformal metric on C^2}). By Lemma \ref{self similar iff minimal wrt g}, $F_\infty$ is then a Lagrangian branched conformal self-shrinker.
\end{proof}

\section{Proof of Theorem \ref{main thm 2}}
\begin{proof} Let $\{F_n\}$ be a sequence of immersed Lagrangian self-shrinking tori in $\mathbb C^2$ with a uniform area upper bound. Let $h_n$ be the metric on the torus $T$ which is conformal to $F_n^*\il \cdot, \cdot\li$ and with zero Gauss curvature. If we can show that $h_n$ stays in a bounded domain of the Teichm\"uller space, then (i) follows from Theorem \ref{main theorem}.

Using Lemma \ref{F in U}, Lemma \ref{U embed in N} and Lemma \ref{self similar iff minimal wrt g}, we also treat each $F_n$ as a minimal immersion in $N$, this means that $F_n:(T, h_n)\to (N,\bar{g})$ is conformal and harmonic. Assume the contrary that the conformal structures degenerate. In this case, there is a mapping $\hat{F}_\infty$ from $\Sigma_\infty$ to $N$  and the  image $F_n(T)$ converges in the Hausdorff distance to $\hat F_\infty(\Sigma_\infty)$ in $N$. Here $\Sigma_\infty$ is a stratified surfaces $\Sigma_\infty = \Sigma_0 \cup \Sigma_b$ formed by the {\it principal component} $\Sigma_0$ and {\it bubble component} $\Sigma_b$. The principal component $\Sigma_0$ is formed by pinching several closed, homotopically nontrivial curves in $T$ and the bubbling component is a union of spheres. There are no necks between the components since $F_n$ is conformal. The map $\hat F_\infty$ is continuous on $\Sigma_\infty$  and harmonic when restricted to each component of $\Sigma_\infty$. Since all the components intersect each others possibly at finitely many points, $\hat F_\infty$ is harmonic except at a finite set  $\hat S$.

Since the conformal structures determined by the metrics $h_n$ degenerate, at least one homotopically nontrivial closed curve must be pinched to a point as $n\to\infty$. It follows that $\Sigma_0$ is a finite union of $\mathbb S^2$'s. Each of these 2-spheres is obtained by adding finitely many points to the cylinder ${\mathbb S}^1\times{\mathbb R}$ that comes from pinching one or two homotopically nontrivial loops: two at the infinity and at most finitely many at the blowup points of the sequence $F_n$, by the removable singularity theorem of Sacks-Uhlenbeck \cite{SU}. Therefore, $\hat F_\infty$ is a finite union of harmonic mappings $\hat F_\infty^i$ from the sphere to $N$.

Since $F_n$ are conformal, there are no necks between the components. The bubble tree convergence described above are given by the results in \cite{CT} or \cite{CLi}. In \cite{CT}, the limiting surface is a stratified surface with geodesics connecting the two dimensional components. But together with conformality of each $F_n$ and Proposition 2.6 in \cite{CT}, one sees that all the geodesics involved have zero length. Alternatively, we can use the compactness theorem in \cite{CLi} which says: Suppose that $\{f_k\}$ is a sequence of $W^{2,2}$  branched conformal immersions of $(\Sigma, h_k)$ in a compact manifold $M$. If
$$
\sup_k\left\{\mu(f_k)+W(f_k)\right\}<+\infty
$$
then  either $\{f_k\}$ converges to a point, or
there is a stratified sphere $\Sigma_\infty$ and a $W^{2,2}$ branched conformal immersion
$f_\infty:\Sigma_\infty\to M$,
such that a subsequence of
$\{f_k(\Sigma)\}$ converges to  $f_\infty(\Sigma_\infty)$
in the Hausdorff topology, and the area and the Willmore energy satisfy
$$
\mu(f_\infty)=\lim_{k\rightarrow+\infty}\mu(f_k)\,\, \mbox{and} \,\, W(f_0)\leq\lim_{k\rightarrow+\infty} W(f_k).
$$
 The conditions of the theorem are satisfied by the sequence $\{F_n\}$ as the area $\tilde \mu (F_n)$ and the Willmore energy $W(F_n)$ in $N$ (which is zero as each $F_n$ is minimal immersion in $N$) are uniformly bounded. In our situation, $F_k$ will not converges to a point since $\text{diam}(F_n(T)) \geq \sqrt 2$ by equations (\ref{bound in diam}) and (\ref{W = A}). Thus the inequality on the Willmore energies in $N$
$$
W(\hat F_\infty)\leq\lim_{n\to\infty}W(F_n) = 0
$$
implies that the limiting $\mathbb S^2$'s are all branched minimal surfaces in $N$.

Consequently, the images $F_n(T)$ converge in the Hausdorff distance to the image of finitely many harmonic maps $\mathbb S^2 \to N$. These harmonic maps are conformal branched immersions, which are also Lagrangian by similar reasons as in the proof of Theorem \ref{main theorem}. By Theorem \ref{no S2 lag shrinker}, all these harmonic maps are trivial. Hence, the images $F_n(T)$ converge in the Hausdorff distance to a point in $\mathbb C^2$. Again, this is impossible by the diameter estimate $\text{diam}(F_n(T)) \geq \sqrt 2$. This contradiction shows that the conformal structures cannot degenerate and that finishes the proof of (i).

To show (ii), we assume $F_n\to F_\infty$ smoothly by (i). Note that when the area upper bound $\Lambda<16\pi$, by equation (\ref{W = A}), the Willmore energy ${\mathcal W}(F_n)$ of $F_n$ in $({\mathbb R}^4, \langle \cdot,\cdot\rangle)$ admits an upper bound $\Lambda/2<8\pi$. By Theorem 1 in \cite{CLi},
$$
{\mathcal W}(F_\infty)\leq \lim_{n\to\infty}{\mathcal W}(F_n),
$$
thus the limiting Lagrangian shrinker $F_\infty$ also has Willmore energy strictly less than $8\pi$. By Proposition 4.1 in \cite{KLi} (also, Theorem 3.1 there), the limiting mapping $F_\infty: \Sigma \to \mathbb C^2$ has no branch points, therefore it is an immersion. By Theorem 6 in \cite{LY}, the immersion $F_\infty$ must be an embedding.
\end{proof}

\bibliographystyle{amsplain}


TITLE: Approximation theorems
QUESTION [14 upvotes]: The Weierstrass' approximation theorem for continuous functions on a compact space by using polynomials is well-known. As far as I know, there are some variants of this theorem, e.g. Stone-Weierstrass that refers not only to polynomials as approximator functions. Where could I find these Weierstrass-like approximation theorems? On-line references are OK, but one might also point to some books.
Thanks in advance,
Lucian

REPLY [2 votes]: If you want some technical challenge (or let's say, it really is for me) you can have a look at A. Pinkus. N-widths in Approximation Theory, Springer-Verlag, New York, 1980.


TITLE: Version of non-linear functions with linearly increasing arc-length (proof)
QUESTION [1 upvotes]: I am trying to find a function "f" such that: \begin{align}{ \int_{a}^{a+b} \sqrt{(\frac{d}{dt}(x(f(t))))^2+(\frac{d}{dt}(y(f(t))))^2 } \,dt}=B \hspace{3cm}a,b,B\in{\mathbb{R}}\end{align} This is easy for linear functions because the derivatives are constant. I have no idea how to prove how to do it generally. Is there a proof of a way to find the function "f"? I have tried expanding b into the integral of one. \begin{align}\int_{a}^{a+b}\frac{B}{b}dt=B \end{align}I have tried integrating but the integrals always has “f” multiplied by an irreducible function.\begin{align}{  \sqrt{(\frac{d}{dt}(x(f(t))))^2+(\frac{d}{dt}(y(f(t))))^2 } }=1 \end{align}\begin{align}{  (\frac{d}{dt}(x(f(t))))^2+(\frac{d}{dt}(y(f(t))))^2  }=1 \end{align}\begin{align}{ -\sqrt{(\frac{d}{dt}(x(f(t))))^2+(\frac{d}{dt}(y(f(t))))^2 } }\neq1 \end{align} From there I have tried using indefinite integrals but, It just introduces unknowns. I have also tried integrating on the interval [0,t] but it quickly becomes unintegrable because the integrand ends up having the following integral:\begin{align}\int_{f^{-1}(f^{-1}(y^{-1}(0)))}^{f^{-1}(t)}f(t)f^{-1}(t)dt\end{align}The most promising solution is inverting the line integral from [a,t] for f(t)=t. The result has no elementary inverse but, I can use root finding methods then concatenate  over the lines from a to t to create a piecewise equation that is a discrete version of r(f(t)) where r(t)=(x(t),y(t)).Then I can shorten the lengths to find a better approximation. The trouble is proving that it works.
\begin{align}\end{align}Edit: This question is about the reparameterization of parametric functions (in t) such that the change in t equals the change in arclength. The question is how to calculate that reparameterization and a proof that it works.

REPLY [1 votes]: A couple of suggestions …
If you’re working with polynomial (i.e. Bézier) curves, they have a subset called Pythagorean Hodograph curves — the cool kids call them PH curves. On a PH curve, the troublesome square root goes away and the norm of the first derivative vector is a polynomial function. So, it’s easy to integrate to get arclength. Not quite what you were asking, but related, I think. Rida Farouki wrote an entire book about PH curves, if you’re interested.
More generally, presumably you can calculate a few values of $t$ and $s$, and do some simple curve fitting that gives you (approximately) $t$ as a function of $s$, or vice versa.
You asked for a proof that this works. If you do the reparameterisation curve fitting using polynomials, then the Stone-Weierstrass theorem tells you that the approximation can be made as good as you like.


TITLE: Proof of Hardy's Classic Integral Inequality
QUESTION [0 upvotes]: While searching for different proofs of Hardy integral inequality, I saw a proof that used Homogeneity of norm and a kernel function. The first line of the proof says:
Let $$F(x)=\frac{1}{x}\int_{0}^{x}{f(t)dt}= \int_{0}^{1}{f(tx)dt}.$$
The function $F(x)$ is the Hardy mean operator. I am unable to follow the equality of the two integrals above which is mentioned  here in Proof 2 of the following paper https://kkms.org/index.php/kjm/article/download/289/204
Help: If anyone can please help me understand it, I will be very grateful.

REPLY [0 votes]: Let $g(t) = tx$ for a fixed $x$ and note that $g’(t) = x$, which is constant w.r.t. $t$. Then read the equality from the right to the left and use a change of variables  to deduce that
 $$\begin{align}
\int_0^1 f(tx) dt &= \int_0^1 f(g(t)) dt \\
 &= \int_0^1 f(g(t)) \frac{g’(t)}{g’(t)} dt \\ &= \frac 1x \int_0^1 f(g(t)) g’(t) dt \\&= \frac 1x \int_{g(0)}^{g(1)} f(t) dt \\ &= \frac 1x \int_0^x f(t) dt
\end{align}$$
as desired.


TITLE: Is it possible to hear the past?
QUESTION [1 upvotes]: From this Stack Exchange Physics Post, I am certain that it is possible to view the past.
But then this interesting question came to me.
Is it possible to hear the past?
Ok, you might say, "Well, we are hearing the past, aren't we?" and go onto immense details about the speed of sound against the difference and stuff.
But what if I wanted to hear the sound of a place extremely close by days, weeks, months or even years back?
Would that be possible?

REPLY [0 votes]: i think it's possible. since sounds waves never die and it keep moving in the earth forever, maybe we can hear the sound of people who was in the room? but not with our ears maybe using the new technology.
i see it scary somehow. hearing someone's voice and he's dead, but it's cool!


text:
\begin{document}

\title{A Non-differential Distributed  Space-Time Coding for Partially-Coherent Cooperative Communication}
\author{Harshan J.~and
        B. Sundar Rajan,~\IEEEmembership{Senior Member,~IEEE}

\thanks{This work was supported through grants to B.S.~Rajan; partly by the IISc-DRDO program on Advanced Research in Mathematical Engineering, and partly by the Council of Scientific \& Industrial Research (CSIR, India) Research Grant (22(0365)/04/EMR-II). The material in this paper was presented in parts at the IEEE International Conference on Communications (ICC 2008), Beijing, China, May. 19-23, 2008. Harshan J. and B. Sundar Rajan are with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore-560012, India. Email:\{harshan,bsrajan\}@ece.iisc.ernet.in.}
\thanks{Manuscript received May 18, 2007; revised January 18, 2008.}}

\markboth{IEEE Transactions on Wireless Communications ,~Vol.~xx, No.~xx, xxxx}{Harshan \MakeLowercase{and} Rajan: A Non-differential Distributed  Space-Time Coding for Partially-Coherent Cooperative Communication}

\maketitle

\begin{abstract}
In a distributed space-time coding scheme, based on the relay channel model, the relay nodes co-operate to linearly process the transmitted signal from the source and forward them to the destination such that the signal at the destination appears as a space time block code. Recently, a code design criteria for achieving full diversity in a partially-coherent environment have been proposed along with codes based on differential encoding and decoding techniques. For such a set up, in this paper, a non-differential encoding technique and construction of distributed space time block codes from unitary matrix groups at the source and a set of diagonal unitary matrices for the relays are proposed. It is shown that, the performance of our scheme is independent of the choice of unitary matrices at the relays. When the group is cyclic, a necessary and sufficient condition on the generator of the cyclic group to achieve full diversity and to minimize the pairwise error probability is proved. Various choices on the generator of cyclic group to reduce the ML decoding complexity at the destination is presented. It is also shown that, at the source, if non-cyclic abelian unitary matrix groups are used, then full-diversity can not be obtained. The presented scheme is also robust to failure of any subset of relay nodes.
\end{abstract}

\begin{keywords}
Cooperative communication, cyclic groups, distributed space-time codes and unitary space-time codes.
\end{keywords}

\section{Introduction and preliminaries}

Co-operative communication can be based on a relay channel model where a set of distributed antennas belonging to multiple users is exploited for achieving spatial diversity \cite{SEA1}-\cite{JiH1}. In \cite{JiH1}, the idea of Space-Time Coding (STC) devised for point to point co-located multiple antenna systems is applied to a two-hop wireless relay network and is referred as Distributed Space-Time Coding (DSTC). The technique involves a two phase protocol where, in the first phase, the source broadcasts the information to the relays and in the second phase, relays linearly process the received signals and forward them to the destination such that the signal at the destination appears as a Space-Time Block Code (STBC).

In the above technique, the destination may or may not have the knowledge of channel fade coefficients (i) from the source to the relays and (ii) from the relays to the destination when decoding for the source signal. DSTC when the destination does not have the knowledge of the channels from the source to the relays but has the knowledge of the channels from the relays to itself is called partially-coherent DSTC, which is the subject matter of this paper. For a partially-coherent set up, in \cite{KiR1}, a code design criteria is proposed for achieving full diversity.  A differential encoding and decoding strategy with a class of full diversity achieving codes using cyclic division algebras is also proposed. Inspired by non-coherent MIMO unitary differential modulation \cite{HoS}, differential distributed space time codes have been proposed in \cite{JiJ2}, \cite{OgH1} and \cite{RaR}.

In \cite{KiR1}, it is shown that the coding problem for a partially coherent setup is to distributively design a finite set of unitary matrices such that the matrix obtained from juxtaposing any two distinct matrices from the set must be full rank. In other words, the problem is to design non-intersecting subspaces such that the principal angles between subspaces is as large as possible. However, with the use of differential techniques \cite{KiR1,JiJ2,OgH1, RaR}, the coding problem gets transformed in to a problem of constructing a finite set of unitary matrices such that the difference matrix of any two matrices is full rank. This is a well known design criteria for relay networks with amplify and forward protocol in a coherent detection environment and code constructions based on the above criteria are available in the literature. In this paper, we propose a method to explicitly construct non-intersecting subspaces for a partially-coherent set up. i.e we construct a set of unitary matrices in a distributed way such that the matrix obtained from juxtaposing any two distinct matrices from the set must be full rank.\\
\indent The contributions of this paper can be summarized as follows: (i) for the partially-coherent set up, we introduce a non-differential coding scheme and obtain an expression for ML decoding metric (Theorem \ref{thm1}) for the general class of unitary Distributed Space-Time Block Codes (DSTBCs) (See Definition \ref{udstbc}). (ii) We construct unitary DSTBCs using cyclic unitary matrix groups called Non-differential Cyclic Distributed Space-Time Codes (NCDSTC) (Definition \ref{cdstbc}) and provide a necessary and sufficient condition on its generator such that the NCDSTC is fully diverse (Theorem \ref{thm2}) and the pairwise error probability (PEP) is minimized. (iii) Also, we provide conditions on the choice of the generator of the cyclic group so as to reduce the decoding complexity at the destination. (iv) We show that the proposed scheme is robust to failure of a subset of the relay nodes.

The remaining part of the paper is organized as follows: In Section \ref{sec2}, along with the signal model, partially-coherent distributed space time coding technique is briefly reviewed and an ML decoding metric for a class of unitary DSTBCs is presented. The scheme of non-differential unitary DSTC from cyclic unitary matrix groups is introduced in Section \ref{sec3} where we provide a necessary and sufficient condition on the generator of the cyclic group to achieve full diversity and to minimize the PEP. In Section \ref{sec4}, an ML decoding metric for the proposed codes is presented along with two reduced decoding complexity ML decoders. Details on the robustness of our scheme to the failure of a subset of the relay nodes is also presented. Possible directions of future work and concluding remarks constitute Section \ref{sec5}.\\

\noindent
\textit{Notations:} For a complex matrix $\textbf{X}$, the matrices $\textbf{X}^*$, $\textbf{X}^T$,  $\textbf{X}^{H}$, $|\textbf{X}|$, $\mbox{Re } \textbf{X}$ and $\mbox{Im }\textbf{X}$ denote, respectively, the conjugate, transpose, conjugate transpose, determinant, real part and imaginary part of $\textbf{X}$.
For a complex matrix $\textbf{Y}$ of the order same as $\textbf{X}$, $\textbf{X} \odot \textbf{Y}$ denotes the Hadamard product of $\textbf{X}$ and $\textbf{Y}$. The $T\times T$ identity matrix is denoted by $\textbf{I}_T$  and $\textbf{O}_T$ denotes the $T$-length vector of zeros.  We use $|x|$ to denote the absolute value of the complex number $x$ and $E \left[x\right]$ to denote the expectation of the random variable $x.$ We write $\textbf{x} \sim \mathcal{CSCG} \left(\mu, \mathbf{\Gamma} \right) $ when $\textbf{x}$ is a circularly symmetric complex Gaussian random vector with mean $\mu$ and covariance matrix $\mathbf{\Gamma}$ and use $j$ for $\sqrt{-1}.$ The set of integers and the set of complex numbers are, respectively, denoted by ${\mathbb Z}$ and ${\mathbb C}.$

\section{Partially Coherent Distributed space time coding}
\label{sec2}

\begin{figure}
\centering
\includegraphics[width= 2.7in]{rajan_paper-tw-may-07-0526_fig1}
\caption{Wireless relay network model}
\label{model}
\end{figure}

\subsection{Signal model}
The wireless network considered in Figure \ref{model} consists of $R + 2$ nodes each having single antenna which are placed randomly and independently according to some distribution. There is one source node and one destination node. All the other $R$ nodes work as relays. We denote the channel from the source node to the $j$-th relay as $f_{j}$ and the channel from the $j$-th relay to the destination node as $g_{j}$ for $j=1,2, \cdots, R$.
\noindent The following assumptions are made in our system model: (i) all the nodes are subjected to half duplex constraint. (ii) fading coefficients  $f_{j},g_{j}$ are i.i.d $ \mathcal{CSCG} \left(0,1 \right)$ with coherence time interval, $T.$ (iii) all the nodes are synchronized at the symbol level. (iv) Destination knows the fading coefficients $g_{j}$'s but not $f_{j}$'s.\\

\indent Every transmission from the source to the destination comprises of two phases. In the first phase the source transmits a $T$ length complex vector from the codebook $\mathcal{S}$ = $\left\{ \textbf{s}_{1},\, \textbf{s}_{2},\, \textbf{s}_{3},\, \cdots , \textbf{s}_{L} \right\} $ consisting of information vectors $\textbf{s}_{l} \in \mathbb{C}^{T}$ such that $E\left[\textbf{s}_{l}^{H}\textbf{s}_{l}\right]$ = 1, so that $P_{1}T$ is the average transmit power. In particular, $P_{1}$ is the average transmit power used at the source node for every channel use. When the information vector $\textbf{s}$ is transmitted, the received vector at the $j$-th relay is given by $\textbf{r}_{j} = \sqrt{P_{1}T}f_{j}\textbf{s} + \textbf{n}_{j},  \textit{ j} = 1,2,\cdots, R$. In the second phase, all the relay nodes are scheduled to transmit $T$ length vectors to the destination simultaneously. Each relay is equipped with a fixed  $T\times T$  unitary matrix $\textbf{A}_{j}$ and is allowed to linearly process the received vector. The $j^{th}$ relay is scheduled to transmit $\textbf{t}_{j} = \sqrt{\frac{P_{2}}{(1 + P_{1})}}\textbf{A}_{j}\textbf{r}_{j}$ where $P_{2}$ is the average transmit power used at each relay for every channel use.
\noindent
The vector received at the destination is given by
\begin{equation}\label{bfy}
\textbf{y} = \sum_{j = 1}^{R} (g_{j}\textbf{t}_{j}) + \textbf{w} = \sqrt{\frac{P_{1}P_{2}T}{(1 + P_{1})}}\textbf{S}\textbf{h} + \textbf{n}
\end{equation}

\noindent where $\textbf{w} \sim \mathcal{CSCG} \left(0, \textbf{I}_{T} \right)$ is the additive noise at the destination,
$\textbf{n} = \sqrt{\frac{P_2}{(1 + P_1)}}\left[ \sum_{j=1}^{R} (g_{j}\textbf{A}_{j}\textbf{n}_{j}) \right]  + \textbf{w} \label{bfN}.$ The equivalent channel  \textbf{h} is given by $\textbf{h} = [f_{1}g_{1} ~ f_{2}g_{2} ~ \cdots ~ f_{R}g_{R} ]^T \in \mathbb{C}^{R}$. The codeword matrix $\textbf{S}$ is given by $\textbf{S} = \left[  \textbf{A}_{1}\textbf{s}~~  \textbf{A}_{2}\textbf{s}~~  \cdots ~~ \textbf{A}_{R}\textbf{s} \right] \in {\mathbb C}^{T \times R}.$

\begin{definition}
\label{dstbc}
The collection $\mathcal{C} = \left\{ \left[  \textbf{A}_{1}\textbf{s}~~  \textbf{A}_{2}\textbf{s}~~  \cdots ~~ \textbf{A}_{R}\textbf{s} \right]\right\}$ of codeword matrices when $\textbf{s}$ runs over $\mathcal{S}$, is called the Distributed Space-Time Block code (DSTBC).
\end{definition}

\subsection{ML Decoder}
Since the relay matrices $\textbf{A}_{j}$ are unitary, the random vectors $\textbf{w}$ and $\textbf{n}_{j}$ are independent and Gaussian and since $g_{j}$ are known at the receiver, $\textbf{n}$ is a Gaussian random vector with
\begin{center}
 $E\left[ \textbf{n}\right]$  = $\textbf{0}_{T}$ ~ and
\end{center}
\begin{center}
 $E\left[ \textbf{n}\textbf{n}^{H}\right]$   = $\left(1 + \frac{P_{2}}{(1 + P_{1})}\sum_{j = 1}^{R}(|g_{j}|^{2})\right)\textbf{I}_T.$
\end{center}
Assume that $\textbf{S} \in \mathcal{C}$. When the destination has the knowledge of $g_{j}$'s but not of $f_{j}$'s, ${\textbf y}$ is a Gaussian random vector with

\begin{center}
$E\left[{\textbf y}|\textbf{S} , g_{j} \right] = \textbf{0}_{T}$ ~ and
\end{center}

\begin{center}
$E\left[ \textbf{yy}^{H}| \textbf{S}, g_{j} \right] = \mathbf{\Sigma}_{y} = \rho \textbf{S} \mathbf{\Sigma}_{h}\textbf{S}^{H} +  \gamma \textbf{I}_T$
\end{center}
where,
\begin{itemize}
\item $\rho = \frac{P_{1}P_{2}T}{P_{1} + 1}$
\item $\gamma = (1 + \frac{P_{2}}{(1 + P_{1})}\sum_{j = 1}^{R}(|g_{j}|^{2}))$ ~ and
\item $\mathbf{\Sigma}_{h} = \mbox{diag}\left( |g_{1}|^{2},\cdots |g_{R}|^{2}\right).$
\end{itemize}
Then, the conditional pdf $P(\textbf{y}| \textbf{S} , g_{j})$ is given by

\begin{equation*}
P(\textbf{y}| \textbf{S} , g_{j}) =\frac{1}{|\mathbf{\Sigma}_{y}|} exp\left( -\left(\textbf y^{H}\mathbf{\Sigma}_{y}^{-1}\textbf y\right)\right) .
\end{equation*}
The partially-coherent ML decoder decodes to a codeword $\hat{\textbf{S}}$ where
\begin{equation}
\label{ML_initial}
\hat{\textbf{S}} = arg\, \max_{\textbf{S} \in \mathcal{C}} P(\textbf{y}|\textbf{S} , g_{j}).\\
\end{equation}

\begin{definition}
\label{udstbc}
\cite{KiR1} A DSTBC is called a unitary DSTBC if every $\textbf{S} \in \mathcal{C}$ in Definition \ref{dstbc} satisfies the condition that $\textbf{S}^{H}\textbf{S} = t \textbf{I}_{R}$, where $t$ is a constant real number independent of the codeword $\textbf{S}$.
\end{definition}

\begin{theorem}
\label{thm1}
For a unitary DSTBC, the partially-coherent ML decoding is given by
\begin{equation}
\label{ML}
arg\, \max_{\textbf{S} \in~\mathcal{C}} \left(\textbf{y}^{H}\textbf{S}\textbf{G}\textbf{S}^{H}\textbf{y}\right),
\end{equation}
\begin{equation*}
\label{G_matrix}
~  \mbox{ where } ~ \textbf{G}  = \mbox{diag}\left( \beta_{1} ,\cdots \beta_{R} \right) ~ \mbox{ and }
\end{equation*}
$$\beta_{j} = |g_{j}|^{2}\left(|g_{j}|^{2} + \gamma \rho^{-1} \right)^{-1}~ \mbox{for} ~j = 1, \cdots , R.$$

\end{theorem}

\begin{proof}
Using the well known result, $|\textbf{I} + \textbf{A}\textbf{B}| = |\textbf{I} + \textbf{BA}| $ and the appropriate use of the matrix inversion lemma in \eqref{ML_initial}, the result follows.
\end{proof}

\subsection{Design criteria}
Chernoff bound on the PEP for the decoder in \eqref{ML_initial} is given in \cite{KiR1} following which a  criteria for designing unitary DSTBCs in order to minimize the PEP is also provided. For achieving a diversity order of $R$, the term $|\textbf{S}_{nm}^{H}\textbf{S}_{nm}|$ has to be non zero for all $m$, $n$ such that $n \neq m,$ where $\textbf{S}_{nm} \in \mathbb{C}^{T \times 2R}$ is obtained by juxtaposing two codewords $\textbf{S}_{n}$ and $\textbf{S}_{m}$ as,
\begin{equation}
\label{concate}
\textbf{S}_{nm}= \left[\textbf{S}_{n} ~~ \textbf{S}_{m} \right].
\end{equation}
Further, for large values of $P$ (where $P = P_{1} + RP_{2}$ is the total power used at all nodes for every channel use), PEP can be minimised by maximising $|\textbf{S}_{nm}^{H}\textbf{S}_{nm}| ~ \mbox{ for  all }  n \neq m.$ A necessary condition for this is $T \geq 2R$. Hence, in the rest of the paper, we consider $T = 2R.$\\
\indent In the following section, we construct codes that satisfy the above design criterion for full-diversity using Generalized Butson-Hadamard matrices defined as follows:
\begin{definition}
A Generalized Butson-Hadamard (GBH) matrix \cite{Hor} is a $T \times T$ matrix $\textbf{M}$ with entries such that
\begin{equation*}
\textbf{M}\textbf{M}^{H} = \textbf{M}^{H}\textbf{M} = T\textbf{I}_{T}
\end{equation*}
and the conjugate of every entry  $m_{ij}$ of $\textbf{M}$ is its inverse $m_{ij}^{-1}$ i.e $m_{ij}^{*} = m_{ij}^{-1}.$
\end{definition}

\section{A non-differential scheme and codes using cyclic unitary matrix groups}
\label{sec3}
We introduce a non-differential encoding scheme using a cyclic unitary matrix group \cite{HoS} of diagonal matrices at the source and relay matrices constructed from GBH matrices \cite{OgH1}, \cite{Hor}. Appropriate ingredients required at the source and the relays are as follows:

\subsection{At the Source}
The source node is equipped with a finite abelian group $\mathcal{U}$ of size $L$ consisting of $2R \times 2R$ diagonal unitary matrices and a $2R \times 1$ complex vector, $\textbf{x}$. Let the group $\mathcal{U}$ be written as a direct product of $K$ cyclic groups $\mathcal{U}_{\nu}$, each of order $L_{\nu}$ for $\nu = 1, 2, \cdots, K$ as,
\begin{equation}
\label{group_U}
\mathcal{U} = \mathcal{U}_{1} \times \mathcal{U}_{2} \times \cdots \mathcal{U}_{K}
\end{equation}
where $L = \prod_{\nu = 1}^{K} L_{\nu}.$ When the set of $L_{\nu}$'s for $\nu = 1, 2, \cdots, K$ are pairwise relatively prime, the group $\mathcal{U}$ will be cyclic. Using the mixed-radix notation $l = \left( l_{1}, l_{2}, \cdots, l_{K}\right)$, where each $1 \leq l_{\nu} \leq L_{\nu}$, we denote every element of $\mathcal{U}$ as
\begin{equation*}
\textbf{D}^{l} = \prod_{\nu = 1}^{K} \textbf{D}_{\nu}^{l_{\nu}}
\end{equation*}
where $\textbf{D}_{\nu}$ is a generator of the $\nu^{th}$ cyclic group $\mathcal{U}_{\nu}$  given by
{\footnotesize
\begin{equation}
\label{generators}
\textbf{D}_{\nu} =  \mbox{diag}\left\lbrace exp\left(\frac{j2\pi u_{1 \nu}}{L_{\nu}}\right), ~ exp\left(\frac{j2\pi u_{2 \nu}}{L_{\nu}}\right) \cdots ~ exp\left(\frac{j2\pi u_{2R \nu}}{L_{\nu}}\right) \right\rbrace.
\end{equation}
}

\noindent For each $\nu$, $u_{i \nu} \in \left\lbrace 1, 2,\cdots L_{\nu}-1 \right\rbrace$ for $i = 1, 2, \cdots, 2R$ have to be chosen appropriately to achieve full-diversity and to maximise coding gain.
\noindent The $i$-th diagonal element of $\textbf{D}^{l}$ is given by
\begin{equation*}
\exp\left( j 2 \pi \sum_{\nu = 1}^{K} \frac{u_{i \nu}l_{\nu}}{L_{\nu}}\right).
\end{equation*}
Hence, the source is equipped with a code $\{\mathcal{U}, \textbf{x} \}$ where $\textbf{x} = \frac{1}{\sqrt{2R}} [1~1~1~\cdots~1]^T.$

\subsection{At the relays}
\begin{figure*}
\begin{equation}
\label{zth element}
\textbf{V}^{z}_{12} = R\left[ exp\left(j 2\pi \sum_{\nu = 1}^{K} \frac{l_{\nu}(u_{z\nu}-u_{(R + z)\nu})}{L_{\nu}}\right) + exp\left(j 2\pi \sum_{\nu = 1}^{K} \frac{\hat{l}_{\nu}(u_{z\nu}-u_{(R + z)\nu})}{L_{\nu}}\right) \right]
\end{equation}
\hrule
\end{figure*}
Let $\textbf{M}$ be the $R \times R$ GBH matrix, using which we construct a $2R \times R$ matrix $\mathbf{\Gamma}$ as $\mathbf{\Gamma} = \left[ \textbf{M}^{T} ~ \textbf{M}^{T} \right]^{T}$. The columns of $\mathbf{\Gamma}$ are used to construct diagonal unitary matrices for the relays as $\textbf{A}_{j} = \mbox{diag} \left( \mathbf{\Gamma}_{1j}, \mathbf{\Gamma}_{2j}, \cdots \mathbf{\Gamma}_{2Rj} \right)$ where $\mathbf{\Gamma}_{ij}$ is the $(ij)-$th element of $\mathbf{\Gamma}$ and $j=1,2, \cdots, R.$

\subsection{Encoding Scheme}
The source node maps $\mbox{log}_{2}L$ bits of information on to one of the $L$ matrices from $\mathcal{U}$ say, $\textbf{D}^{k}$ for some $k$ where $1 \leqslant k \leqslant  L$ and transmits the $2R \times 1$ vector $\textbf{D}^{k}\mathbf{x}$ to all the relays. Each relay performs linear processing on its received vector using the unitary matrix $\textbf{A}_{j}$ and transmits a $2R$ length vectors to the destination. The distributed space-time codeword is of the form,
\begin{eqnarray}
\label{ncdstc_codeword}
\textbf{S}_{k} &=& \left[ \textbf{A}_{1}\textbf{D}^{k}\textbf{x} ~~ \textbf{A}_{2}\textbf{D}^{k}\textbf{x} ~\cdots ~\textbf{A}_{R}\textbf{D}^{k}\textbf{x} \right] \\
 &=& \textbf{D}^{k}\left[ \textbf{A}_{1}\textbf{x} ~~ \textbf{A}_{2}\textbf{x} ~\cdots ~\textbf{A}_{R}\textbf{x} \right] \nonumber \\
 &=&  \frac{1}{\sqrt{2R}}\textbf{D}^{k}\mathbf{\Gamma} \nonumber
\end{eqnarray}
\noindent
where we have used the fact that $\textbf{D}^k$ and $\textbf{A}_j$ commute for all $j.$ It can be observed that $\textbf{S}_{k}^{H}\textbf{S}_{k} = \textbf{I}_{R}$.
\begin{definition}
\label{cdstbc}
The collection $\mathcal{C}$ of $2R \times R$ codeword matrices shown below when $k$ runs from 1 to L
\begin{equation}
\label{code}
\mathcal{C} = \left\lbrace \frac{1}{\sqrt{2R}}\textbf{D}^{k}\mathbf{\Gamma} \right\rbrace
\end{equation}
is called a Non-differential Abelian DSTBC (NADSTBC) and if the group is cyclic, then it is called a Non-differential Cyclic Distributed Space-Time Block Code (NCDSTBC).
\end{definition}

\subsection{Design criteria for $\textbf{D}_{\nu}$}
In this subsection, we derive a criteria for choosing the generators of each of $\mathcal{U}_{\nu}$ such that a NADSTBC is fully diverse.

\begin{theorem}
\label{thm2}
The diversity order of a wireless relay network with $R$ relays using a NADSTBC is $R$ if and only if the generators $\textbf{D}_{\nu}$ in \eqref{generators} are chosen such that, for each $i = 1, 2, \cdots R,$
\begin{equation}
\label{condition_abelian}
\sum_{\nu = 1}^{K} \frac{(l_{\nu} - \hat{l}_{\nu}) (u_{i\nu}-u_{(R + i)\nu})}{L_{\nu}} \notin \mathbb{Z}
\end{equation}
for all pair $(l, \hat{l})$ such that $l \neq \hat{l}.$
\end{theorem}

\begin{proof} Let $\frac{1}{\sqrt{2R}}\textbf{D}^{l}\mathbf{\Gamma}$, $\frac{1}{\sqrt{2R}}\textbf{D}^{\hat{l}}\mathbf{\Gamma} \in \mathcal{C}$. The $2R \times 2R$ matrix as in \eqref{concate} is given by,
\begin{equation*}
\textbf{S}_{l \hat{l}}= \frac{1}{\sqrt{2R}}\left[ \textbf{D}^{l}\mathbf{\Gamma} ~ \textbf{D}^{\hat{l}}\mathbf{\Gamma}\right].
\end{equation*}

\noindent Since $|\textbf{S}^{H}_{l \hat{l}}\textbf{S}_{l \hat{l}}| = |\textbf{S}_{l \hat{l}}\textbf{S}^{H}_{l \hat{l}}|$, we consider the choice of $\textbf{D}_{\nu}$ for $\nu = 1, 2, \cdots K$ such that $|\textbf{S}_{l \hat{l}}\textbf{S}^{H}_{l \hat{l}}| \neq 0$ where

\begin{equation}
\label{justopse}
\textbf{S}_{l \hat{l}}\textbf{S}^{H}_{l \hat{l}} = \frac{1}{2R}\left[\begin{array}{cc}
\textbf{V}_{11}  & \textbf{V}_{12}\\
\textbf{V}_{21} & \textbf{V}_{22}\\
\end{array}\right]\\
\end{equation}
with $\textbf{V}_{11} , \textbf{V}_{12} ,\textbf{V}_{21}$ and $\textbf{V}_{22}$ are $R \times R$ diagonal matrices given by
\begin{equation*}
\textbf{V}_{11} = 2R~\textbf{I}_{R \times R};~~
\textbf{V}_{21} = \textbf{V}_{12}^{H};~~
\textbf{V}_{22} = 2R~\textbf{I}_{R \times R}
\end{equation*}
and for each $z$ from $1, 2, \cdots, R$ the $z$-th diagonal entry of $\textbf{V}_{12}$ is given in \eqref{zth element}.
\noindent
\noindent Applying the result,
{\small
\begin{equation*}
\left |\left[\begin{array}{rr}
\textbf{A}  & \textbf{B}\\
\textbf{C} & \textbf{D}\\
\end{array}\right]\right | = |\textbf{A}|| \textbf{D} - \textbf{C}\textbf{A}^{-1}\textbf{B}|
\end{equation*}
}

\noindent where $\textbf{A} , \textbf{B} , \textbf{C}$ and $\textbf{D}$ are square matrices of same order and  $\textbf{A}^{-1}$ exists, on the matrix in \eqref{justopse} and using \eqref{zth element}, we get
\begin{equation*}
|\textbf{S}_{\hat{l}l}\textbf{S}_{\hat{l}l}^{H}|  = \left(\frac{1}{2R}\right)^{2R}|\textbf{V}_{11}||\textbf{V}_{22} - \textbf{V}_{21}\textbf{V}_{11}^{-1}\textbf{V}_{12}|
\end{equation*}
\begin{equation*}
 = \left(\frac{1}{2R}\right)^{R} |\mbox{diag}\left( \delta_{1}~ \delta_{2} ~\cdots ~\delta_{R} \right)|
\end{equation*}
where,
{\small
\begin{eqnarray*}
\delta_{i} = R\left[1 - \cos\left(2\pi \sum_{\nu = 1}^{K} \frac{(l_{\nu} - \hat{l}_{\nu}) (u_{i\nu}-u_{(R + i)\nu})}{L_{\nu}}\right)\right]
\end{eqnarray*}
}

\noindent For $|\textbf{S}_{l \hat{l}}\textbf{S}^{H}_{l \hat{l}}|$ not to be zero, the product $\delta_{1} \delta_{2} \cdots \delta_{R}$ must be a nonzero value. For each $\delta_{i}$ not to be zero, $u_{i\nu}$ and $u_{(i + R)\nu}$ for all $\nu = 1, 2, \cdots, K$ has to be chosen such that
\begin{equation*}
\sum_{\nu = 1}^{K} \frac{(l_{\nu} - \hat{l}_{\nu}) (u_{i\nu}-u_{(R + i)\nu})}{L_{\nu}} \notin \mathbb{Z}
\end{equation*}
for all $l,\hat{l}$ such that $l \neq \hat{l}.$ It is clear that the above condition on $\textbf{D}_{\nu}$ is both sufficient and necessary.
\end{proof}

\begin{corollary}
\label{coral2}
DSTBCs from non-cyclic abelian groups are not fully diverse for any choice of $\textbf{D}_{\nu}$.
\end{corollary}
\begin{proof}
Since each of the cyclic groups $\mathcal{U}_{\nu} \subseteq \mathcal{U}$, from the special case of Theorem \ref{thm2} with $K=1,$ every $\textbf{D}_{\nu}$ must satisfy the criteria,
\begin{equation*}
\gcd\left( u_{\nu i} - u_{\nu (i + R)}, L_{\nu}\right)  = 1  \mbox{ for } i = 1, \cdots R.
\end{equation*}
Since at least two of the $L_{\nu}$ share a factor and each $\textbf{D}_{\nu}$ satisfy the above criteria, condition in \eqref{condition_abelian} of Theorem \ref{thm2} will not be satisfied for at least one pair of values  $l$ and $\hat{l}$. Hence, non-cyclic abelian DSTBCs cannot provide full diversity when used in the proposed scheme.
\end{proof}

Henceforth, throughout the paper, we will consider only NCDSTBCs. We explicitly state Theorem \ref{thm2} for NCDSTBC as a corollary as:
\begin{corollary}
\label{coral1}
Let $\mathcal{U}$ in \eqref{group_U} be a cyclic group with a generator $\textbf{D}$ given by
\begin{equation*}
\textbf{D} = \mbox{diag}\left\lbrace exp\left(j 2 \pi \frac{u_{1}}{L}\right), exp\left(j 2 \pi \frac{u_{2}}{L}\right) \cdots exp\left(j 2 \pi \frac{u_{2R}}{L}\right) \right\rbrace.
\end{equation*}
A necessary and sufficient condition on the choice of $\textbf{D}$ such the that diversity order of the wireless relay network is $R$, is given by
\begin{equation*}
\gcd(u_{i} - u_{R + i}, L ) = 1 \mbox{ for each } i = 1, 2 , \cdots R.
\end{equation*}
\end{corollary}
\begin{proof}
The result follows as a special case of Theorem \ref{thm2} for $K$ = 1.
\end{proof}

Specializing the proof of Theorem \ref{thm2} for the cyclic group case, the coding gain for NCDSTBC is given by
\begin{equation}
\label{coding_gain_metric}
\phi =  \min_{1 \leqslant l , \hat{l} \leqslant  L, l \neq \hat{l} }\prod_{i = 1}^{R}\left[1 - \cos\left(\frac{2\pi(l-\hat{l})(u_{i} - u_{i + R})}{L}\right)\right].
\end{equation}
\noindent Thus, the vector $\textbf{v}$ has to be chosen such that $\phi$ is maximized, where
\begin{equation}
\label{v-vector}
\textbf{v} = \left( u_{1}-u_{R +1}, u_{2}-u_{R +2}  \cdots u_{R}-u_{2R}\right).
\end{equation}

From the result of Theorem \ref{thm2} and Corollary \ref{coral1}, the performance of NCDSTBC is characterized by \eqref{coding_gain_metric} which is independent of the unitary matrices $\textbf{A}_{j}$ at the relays and  dependent only on the generator of the cyclic group.

\subsection{On the choice of $\mathbf{\Gamma}$}
Instead of a GBH matrix, $\textbf{M}$ can also be any $R \times R$ unitary matrix with no zero entry, using which $\mathbf{\Gamma}$ is constructed as
\begin{equation*}
\mathbf{\Gamma} = \sqrt{R}\left[\begin{array}{r}
\textbf{M}\\
\textbf{M}\\
\end{array}\right].
\end{equation*}
Though, the relay matrices $\textbf{A}_{j}$ are not unitary in this case, simulation results show that, for a given $\mathcal{U}$ at the source, BLER (Block error rate, which corresponds to errors in decoding a codeword is considered as error events of our interest throughout the paper) performance is independent of the above choice of $\textbf{M}$. This can be seen from Figure \ref{GBH_normal-Unitary} which is obtained using the code parameters $L =16$ and $\textbf{v} = [5, 7, 11, 1]$ for a network with 4 relays. However, decoding metric in \eqref{ML} doesn't hold in the latter case. Therefore, we continue to use only GBH matrices to construct $\textbf{A}_{j}$ in the rest of the paper.

\begin{figure}
\centering
\includegraphics[width= 2.9in]{rajan_paper-tw-may-07-0526_fig2}
\caption{Plot illustrating the independence of the BLER on relay specific diagonal matrices}
\label{GBH_normal-Unitary}
\end{figure}

\section{Reduced ML Decoding complexity for NCDSTBCs and Node failures}
\label{sec4}
For a general class of unitary DSTBCs used in a partially-coherent scheme, an ML decoding metric is given in \eqref{ML}. In this section, we specialize this decoding metric for the class of  NCDSTBCs.
\begin{theorem}
\label{thm3}
For a  NCDSTBC in \eqref{code}, the partially-coherent ML decoding given in \eqref{ML} reduces to
\begin{equation}
\label{ML_cdstc}
arg\, \max_{k ~ \in ~ 1, 2, \cdots ~ L} \left(\textbf{y}_{1}^{H} \textbf{G}_{1}\textbf{y}_{1} + \mbox{Re} \left\lbrace{\textbf{y}_{1}^{H}\textbf{G}_{2}\textbf{y}_{2}}\right\rbrace + \textbf{y}_{2}^{H} \textbf{G}_{3}\textbf{y}_{2} \right).\\
\end{equation}
where,
\begin{equation*}
\textbf{y}_{1} = \left[y_{1} ~ y_{2} ~\cdots ~y_{R}\right]^{T},~
\textbf{y}_{2} = \left[y_{R +1} ~ y_{R + 2} \cdots ~y_{2R}\right]^{T} \mbox{ and }
\end{equation*}
$$\textbf{G}_{1} =  \mathbf{\Upsilon}_{1} \odot \mathbf{\Omega}, ~ \textbf{G}_{2} =  \mathbf{\Upsilon}_{2} \odot \mathbf{\Omega} \mbox{  and  } \textbf{G}_{3} =  \mathbf{\Upsilon}_{3} \odot \mathbf{\Omega}$$
where $\odot$ denotes the Hadamard product and $\mathbf{\Upsilon}_{1}, \mathbf{\Upsilon}_{2}$, $\mathbf{\Upsilon}_{3} \in \mathbb{C}^{R \times R}$ given by $\left[ \mathbf{\Upsilon}_{1}\right]_{i, j} = w^{(u_{i} - u_{j})k}$, $\left[ \mathbf{\Upsilon}_{2}\right]_{i, j} = w^{(u_{i} - u_{ R + j})k}$, $\left[ \mathbf{\Upsilon}_{3}\right]_{i, j} = w^{(u_{R + i} - u_{R + j})k}$ and

$\mathbf{\Omega}$ is the $R \times R$ Hermitian matrix, with the $(i,j)-$th element given by
\begin{equation}
\label{new_omega}
\mathbf{\Omega}_{i,j} = \sum_{\lambda = 1}^{R} \beta_{\lambda}\left(m_{i \lambda}m_{j \lambda}^{*} \right) ~ \mbox{for} ~ i, j = 1, \cdots , R
\end{equation}
where $m_{ij}$ are the entries of $\textbf{M}$, the chosen GBH matrix and $\beta_{j} = |g_{j}|^{2}\left(|g_{j}|^{2} + \gamma \rho^{-1} \right)^{-1}~ \mbox{for} ~j = 1, \cdots , R.$
\end{theorem}
\begin{proof} The result follows by substituting \eqref{ncdstc_codeword} in \eqref{ML}.
\end{proof}

\indent In the rest of this section, we present various choices on the vector $\textbf{v}$ in \eqref{v-vector} such that the computations required in decoding using the metric given by \eqref{ML_cdstc} is reduced.
For large value of $P$, performance of the scheme is determined by the choice of the vector $\textbf{v}$. There are several ways of choosing the vector $\textbf{v}$ from various choices of $\textbf{u} = [u_{1}, u_{2} \cdots u_{2R}]$. The optimal choice of $\textbf{v}$ can still be made with appropriate values of $u_{1}, u_{2} \cdots u_{R}$ by keeping
\begin{equation*}
\label{redu_cond_1}
u_{R +1} = u_{R + 2}  = u_{R + 3}  = \cdots = u_{2R}.
\end{equation*}
From the above choice of $v$, $\mathbf{\Upsilon}_{3}$ will be independent of the exponent $k$. Therefore the ML decoding metric in \eqref{ML_cdstc} reduces to
\begin{equation}
\label{ML_ncdstc_red}
arg\, \max_{k ~ \in ~ 1, 2, \cdots ~ L} \left(\textbf{y}_{1}^{\dagger} \textbf{G}_{1}\textbf{y}_{1} + \mbox{Re} \left\lbrace{\textbf{y}_{1}^{\dagger}\textbf{G}_{2}\textbf{y}_{2}}\right\rbrace  \right).\\
\end{equation}
Hence, compared to \eqref{ML_cdstc}, the number of computations required in \eqref{ML_ncdstc_red} to decode a codeword is reduced.\\

\indent The vector $\textbf{v}=(v_1,v_2,\cdots,v_{R})$ can also be chosen to be of the form
\begin{equation*}
\label{red_decod_vector_v}
v_{1} = v_{2} = \cdots = v_{R},
\end{equation*}
by choosing $u_{1} = u_{2} = \cdots = u_{R}$ and $u_{R +1} = u_{R + 2} = \cdots = u_{2R}$ such that $\gcd\left( {u_{1} - u_{R + 1}, L}\right)  = 1$.
In such a case, $\mathbf{\Upsilon}_{1}$ and $\mathbf{\Upsilon}_{3}$ are independent of the exponent $k$ and hence the decoding metric is written as
\begin{equation}
\label{reduced_ML}
arg\, \max_{k ~ \in ~ 1, 2, \cdots ~ L} \mbox{Re}~\left\lbrace{w^{(u_{1} -u_{3})k}\textbf{y}_{1}^{H}\mathbf{\Omega}\textbf{y}_{2}}\right\rbrace.
\end{equation}
In general, in order to estimate the most likely transmitted codeword from the source, the destination needs to perform matrix multiplication of order $R \times R$ according to \eqref{ML_cdstc} for every $k$ from 1 to $L$, where as, in the case of the metric given in \eqref{reduced_ML}, the destination needs to perform complex number multiplication for every $k$ to estimate the most likely transmitted codeword. Hence, the number of computations for decoding a codeword at the destination reduces substantially.

\begin{figure}
\centering
\includegraphics[width= 2.9in]{rajan_paper-tw-may-07-0526_fig5}
\caption{BLER plot for 4 relays with L = 16 for different $\textbf{v}$}
\label{BLER_plot_for_several_vs}
\end{figure}

Figure \ref{BLER_plot_for_several_vs} shows the BLER performance against the total power, $P$ for different choices of generator $\textbf{D}$ with $L$ = 16 for a network with 4 relays. The plot shows that, for a choice of $\textbf{v}$ with $v_{1} = v_{2} = \cdots = v_{R}$, the NCDSTC loses out close to 5-6 db compared to the one with the choice of $\textbf{v}=[11, 7, 3, 1]$. The relay matrices used in all the simulations are $\textbf{A}_{1} = \textbf{I}_{8}$, $\textbf{A}_{2} = \mbox{diag}\left\lbrace 1, -1, 1, - 1, 1, -1, 1, -1\right\rbrace$, $\textbf{A}_{3} = \mbox{diag}\left\lbrace 1, 1, -1, - 1, 1, 1, -1, -1\right\rbrace$, $\mbox{ and }\textbf{A}_{4} = \mbox{diag}\left\lbrace 1, -1, -1, 1, 1, -1, -1, 1\right\rbrace$. The decoding metric given in \eqref{ML} has been used to carry out the simulations. In all our simulations, the total power $P$ has been distributed to the source node and the relay nodes as $P_{1} = \frac{P}{2}$ and $P_{2} = \frac{P}{2R}$.

\subsection{Failure of relay nodes}

In this subsection, we analyze the behavior of the proposed codes when a subset of the relay nodes fail to co-operate with the source i.e when a subset of the relays do not transmit information of the source to the destination. The structure of every codeword $\textbf{S}_{k}$ from the NCDSTC $\mathcal{C}$ is given in Definition \ref{cdstbc}. Similar to codes designed for non-differential non-coherent collocated MIMO systems, the columns of a codeword $\textbf{S}_{k}$ can be viewed as a basis of an $R$-dimensional subspace of $\mathbb{C}^{2R}$. Two codewords satisfy the full diversity condition means that, the $R$-dimensional subspace spanned by columns of each of them are non-intersecting. Therefore, a fully diverse NCDSTC can be viewed as a finite set of non-intersecting $R$- dimensional subspaces in  $\mathbb{C}^{2R}$. From the structure of the code, it is easy to see that, every relay contributes a basis element to each of the subspace (code-word). Let $\hat{\textbf{S}_{k}}$ denote a codeword, after say, $d$ of the $R$ relays fail to co-operate with the source. The columns of a new codeword $\hat{\textbf{S}_{k}}$ span a $(R - d)$ dimensional subspace of $R$-dimensional space spanned by the columns of $\textbf{S}_{k}$. Hence, the new set of subspaces (codewords) remains to be non-intersecting and the new code $\mathcal{\hat{C}}$ will remain to be fully diverse with diversity gain $R-d.$ Figure \ref{relay_failure} shows simulation results on the behavior of BLER when one of the relay fails in a network with 4 relays. The plot is obtained by using a code with parameters $L =16$ and $\textbf{v} = [11, 11, 11, 11]$.

\begin{figure}
\centering
\includegraphics[width = 2.9in]{rajan_paper-tw-may-07-0526_fig3}
\caption{BLER performance of NCDSTC with one relay down}
\label{relay_failure}
\end{figure}

\section{Discussion}
\label{sec5}
We considered the problem of code construction based on non-differential technique for wireless relay networks in a partially-coherent environment. Unitary distributed space time block codes were constructed using a cyclic group of diagonal matrices at the source and diagonal relay matrices constructed from GBH matrices. We have shown that the  BLER performance of the proposed scheme is independent of the relay specific unitary matrices. Further, a necessary and sufficient condition on the choice of the generator of cyclic group is provided for achieving full diversity and to minimize the PEP.  Various choices on the generator of cyclic group to reduce the ML decoding complexity at the destination was provided. The resistance of the proposed scheme to the failure of a subset of the relay nodes is also verified. It was also shown that, using non-cyclic abelian group of diagonal matrices at the source doesn't provide full diversity. An interesting direction for future work is to design non-differential encoding techniques such that the unitary distributed space time block codes can be constructed algebraically.

config: arxiv


TITLE: What is meant by invariant under change of coordinates **to first order**?
QUESTION [1 upvotes]: I am studying elementary Lagrangian mechanics, and I'm a bit confused about the what's meant by invariance of the Lagrangian under change of coordinates to first order.
More specifically, Noether's theorem states

For each symmetry of the Lagrangian, there is a conserved quantity.

the textbook gives the following definition for symmetry:

By 'symmetry', we mean that if the coordinates are changed by some small quantities, then the Lagrangian has no first-order change in these quantities.

Okay, but it does not mention what is meant by no change in first order. Does it mean all the terms are of first order?
For example, consider the Lagrangian $$L={m \over 2}({\dot x}^2+{\dot y}^2)-{k \over 2}(x^2+y^2)$$ and I let the change of coordinates be $$x \to x+ \epsilon y$$ $$y \to y-\epsilon x$$ and get $${dL \over d\epsilon} = \epsilon k(x^2+y^2)$$
so according to the textbook, a symmetry means no change of Lagrangian in first order of $\epsilon$, thus I should expect ${dL\over d\epsilon} =0$, so does this imply that I should evaluate the derivative at $\epsilon =0$, or is my interpretation problematic?

REPLY [2 votes]: The appropriate definition of symmetry uses infinitesimal quantities, not just small quantities. Thus, in terms of your question, the Lagrangian is symmetric if $dL/d\epsilon=0$ at $\epsilon=0$.
In terms of your example (rotation of a 2D harmonic oscillator), we have
$$
L \to (1+\epsilon^2) L = L + \mathcal{O}(\epsilon^2)
$$
Thus to first order in $\epsilon$ $L$ does not change: it's invariant under rotations about the $z$-axis.
So, your interpretation appears correct.


TITLE: Conditional Probability based on intersection of two sets, independent and dependent respectively
QUESTION [2 upvotes]: Given $P(A|B \cap C)$, where $B$ is independent of $A$, and $C$ is dependent of $A$. In this case, how should we calculate the probability?

REPLY [0 votes]: When A and C are dependent on each other but A is independent of B, it simply turns out:
P(A|B∩C)=P(A|C).