prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are the stability conditions for a solution of $-\Delta u = f(u)$?

Read the following text and answer briefly.

Context 1: \section{Introduction and main results}


In this note we are interested in the existence versus non-existence of stable sub- and super-solutions of equations of the form
\begin{equation} \label{eq1}
-div( \omega_1(x) \nabla u ) = \omega_2(x) f(u) \qquad \mbox{in $ {\mathbb{R}}^N$,}
\end{equation} where $f(u)$ is one of the following non-linearities: $e^u$,  $ u^p$ where $ p>1$ and $ -u^{-p}$ where $ p>0$   We assume that $ \omega_1(x)$ and $ \omega_2(x)$, which we call \emph{weights},  are smooth positive functions (we allow $ \omega_2$ to be zero at say a point) and which satisfy various growth conditions at $ \infty$     Recall that we say that a solution $ u $ of $ -\Delta u = f(u)$ in $ {\mathbb{R}}^N$ is stable provided
\[ \int f'(u) \psi^2 \le \int | \nabla \psi|^2, \qquad \forall \psi \in C_c^2,\] where $ C_c^2$ is the set of $ C^2$ functions defined on $ {\mathbb{R}}^N$ with compact support   Note that the stability of $u$ is just saying that the second variation at $u$ of the energy associated with the equation is non-negative      In our setting this becomes:  We say a $C^2$ sub/super-solution $u$ of (\ref{eq1}) is \emph{stable} provided
\begin{equation} \label{stable}
\int \omega_2 f'(u) \psi^2 \le \int \omega_1 | \nabla \psi|^2 \qquad \forall \psi \in C_c^2 
\end{equation}
One should note that (\ref{eq1}) can be re-written as
\begin{equation*}
- \Delta u  + \nabla \gamma(x) \cdot \nabla u ={ \omega_2}/{\omega_1}\ f(u) \qquad \text{ in $ \mathbb{R}^N$},
\end{equation*}
where
$\gamma = - \log( \omega_1)$ and on occasion we shall take this point of view 


\begin{remark} \label{triv} Note that  if $ \omega_1$ has enough integrability then it is immediate that if $u$ is a stable solution  of (\ref{eq1}) we have $ \int \omega_2 f'(u) =0 $  (provided $f$ is increasing)    To see this let $ 0 \le \psi \le 1$ be supported in a ball of radius $2R$ centered at the origin ($B_{2R}$) with $ \psi =1$ on $ B_R$ and such that $  | \nabla \psi | \le \frac{C}{R}$ where $ C>0$ is independent of $ R$    Putting this $ \psi$ into $ (\ref{stable})$ one obtains
\[ \int_{B_R} \omega_2 f'(u) \le \frac{C}{R^2} \int_{R < |x| <2R} \omega_1,\] and so if the right hand side goes to zero as $ R \rightarrow \infty$ we have the desired result 

\end{remark}





The existence versus non-existence of stable solutions of  $ -\Delta u =  f(u)$ in $ {\mathbb{R}}^N$ or $ -\Delta u = g(x) f(u)$ in $ {\mathbb{R}}^N$ is now quite well understood, see  \cite{dancer1, farina1, egg, zz, f2, f3, wei, ces, e1, e2}   We remark that some of these results are examining the case where $ \Delta $ is replaced with $ \Delta_p$ (the $p$-Laplacian) and also in many cases the authors are interested in finite Morse index solutions or solutions which are stable outside a compact set 
  Much of the interest in these Liouville type theorems  stems from the fact that the non-existence of a stable solution is related to the existence of a priori estimates for stable solutions of a related equation on a bounded domain 




       In \cite{Ni}  equations similar to  $ -\Delta u = |x|^\alpha u^p$  where examined on the unit ball in $ {\mathbb{R}}^N$ with zero Dirichlet boundary conditions   There it was shown that for $ \alpha >0$ that  one can obtain positive solutions for $ p $ supercritical with respect to Sobolev embedding and so one can view that the term $ |x|^\alpha$ is restoring some compactness       A similar feature happens for equations of the form
\[ -\Delta u = |x|^\alpha f(u) \qquad \mbox{in $ {\mathbb{R}}^N$};\]    the value of $ \alpha$ can vastly alter the existence versus non-existence of a stable solution,  see \cite{e1, ces, e2, zz, egg} 

We now come to our main results and for this we need to define a few quantities:

\begin{eqnarray*}
I_G&:=& R^{-4t-2} \int_{ R < |x|<2R} \frac{ \omega_1^{2t+1}}{\omega_2^{2t}}dx , \\
 J_G&:=& R^{-2t-1} \int_{R < |x| <2R} \frac{| \nabla \omega_1|^{2t+1} }{\omega_2^{2t}} dx  ,\\I_L&:=& R^\frac{-2(2t+p-1)}{p-1}  \int_{R<|x|<2R }{ \left(   \frac{w_1^{p+2t-1}}{w_2^{2t}}    \right)^{\frac{1}{p-1} }  }  dx,\\ J_L&:= &R^{-\frac{p+2t-1}{p-1} }  \int_{R<|x|<2R }{ \left(   \frac{|\nabla w_1|^{p+2t-1}}{w_2^{2t}}     \right)^{\frac{1}{p-1} }  }  dx,\\
I_M  &:=&   R^{-2\frac{p+2t+1}{p+1} }  \int_{R<|x|<2R }{ \left(   \frac{w_1^{p+2t+1}}{w_2^{2t}}    \right)^{\frac{1}{p+1} }  } \ dx, \\
J_M  &:= &  R^{-\frac{p+2t+1}{p+1} }  \int_{R<|x|<2R }{ \left(   \frac{|\nabla w_1|^{p+2t+1}}{w_2^{2t}}     \right)^{\frac{1}{p+1} }  }  dx 
\end{eqnarray*}


The three equations we examine are
\[ -div( \omega_1 \nabla u ) = \omega_2 e^u \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (G), \]
\[  -div( \omega_1 \nabla u ) = \omega_2 u^p \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (L), \]
\[ -div( \omega_1 \nabla u ) = - \omega_2 u^{-p} \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (M),\]  and where we restrict $(L)$ to the case $ p>1$ and $(M)$ to $ p>0$     By solution we always mean a $C^2$ solution    We now come to our main results in terms of abstract $ \omega_1 $ and $ \omega_2$     We remark that our approach to non-existence of stable solutions is the approach due to Farina, see \cite{f2,f3,farina1} 

\begin{thm} \label{main_non_exist} \begin{enumerate}


\item   There is no  stable sub-solution of $(G)$ if $ I_G, J_G \rightarrow 0$ as $ R \rightarrow \infty$   for some   $0<t<2$ 
 \item    There is no positive   stable sub-solution (super-solution) of $(L)$  if  $ I_L,J_L \rightarrow 0$ as $ R \rightarrow \infty$   for some   $p- \sqrt{p(p-1)}   < t<p+\sqrt{p(p-1)} $  ($0<t<\frac{1}{2}$) 
\item   There is no positive   stable super-solution of (M) if  $ I_M,J_M \rightarrow 0$ as $ R \rightarrow \infty$  for some $0<t<p+\sqrt{p(p+1)}$ 

\end{enumerate}

\end{thm}     If we assume that $ \omega_1$ has some monotonicity we can do  better    We will assume that the monotonicity conditions is satisfied for big $x$  but really all ones needs is for it to be satisfied on a suitable sequence of annuli 

\begin{thm} \label{mono}  \begin{enumerate} \item There is no  stable sub-solution of $(G)$ with $  \nabla \omega_1(x) \cdot  x \le 0$ for big $x$ if  $ I_G \rightarrow 0$ as $ R \rightarrow \infty$   for some   $0<t<2$ 

\item There is no positive  stable sub-solution of  $(L)$ provided  $  I_L \rightarrow 0$ as $ R \rightarrow \infty$ for either:
\begin{itemize}
\item  some $  1 \le t < p + \sqrt{p(p-1)}$ and $  \nabla \omega_1(x) \cdot x \le 0$ for big $x$,  or  \\

\item some $ p - \sqrt{p(p-1)} < t \le 1$ and $ \nabla \omega_1(x) \cdot x \ge 0$ for big $ x$ 
\end{itemize}
 There is no positive super-solution of $(L)$ provided $ I_L \rightarrow 0$ as $ R \rightarrow \infty$ for some $ 0 < t < \frac{1}{2}$ and $ \nabla \omega_1(x) \cdot x \le 0$ for big $x$ 



\item  There is no positive stable super-solution of $(M)$ provided  $ I_M \rightarrow 0$ as $ R \rightarrow \infty$  for some  $0<t<p+\sqrt{p(p+1)}$ 

\end{enumerate}

\end{thm}


\begin{cor}  \label{thing}  Suppose $ \omega_1 \le C  \omega_2$  for big $ x$, $ \omega_2 \in L^\infty$,  $ \nabla \omega_1(x) \cdot  x \le 0$ for big $ x$ 
\begin{enumerate} \item There is no stable sub-solution of $(G)$ if $ N \le 9$ 

\item  There is no positive stable sub-solution of $(L)$ if  $$N<2+\frac{4}{p-1} \left( p+\sqrt{p(p-1)}  \right) $$

\item  There is no positive stable super-solution of $(M)$ if $$N<2+\frac{4}{p+1} \left( p+\sqrt{p(p+1)}  \right) $$

\end{enumerate}

\end{cor}


If one takes $ \omega_1=\omega_2=1$ in the above corollary, the results obtained for $(G)$ and  $(L)$,  and for some values of $p$ in $(M)$, are optimal, see \cite{f2,f3,zz} 





We now drop all monotonicity conditions on $ \omega_1$ 

\begin{cor} \label{po} Suppose  $ \omega_1 \le C \omega_2$ for big $x$, $ \omega_2 \in L^\infty$, $ | \nabla \omega_1| \le C \omega_2$ for big $x$ 
\begin{enumerate} \item  There is no stable sub-solution of $(G)$ if $ N \le 4$ 

\item  There is no positive stable sub-solution of $(L)$ if $$N<1+\frac{2}{p-1} \left( p+\sqrt{p(p-1)}  \right) $$

\item There is no positive super-solution of $(M)$ if $$N<1+\frac{2}{p+1} \left( p+\sqrt{p(p+1)}  \right) $$

\end{enumerate}

\end{cor}

Some of the conditions on $ \omega_i$ in Corollary \ref{po} seem somewhat artificial   If we shift over to the advection equation (and we take $ \omega_1=\omega_2$  for simplicity)
\[ -\Delta u + \nabla \gamma \cdot \nabla u = f(u), \] the conditions on $ \gamma$ become: $ \gamma$ is bounded from below and has a bounded gradient 



In what follows we examine the case where $ \omega_1(x) = (|x|^2 +1)^\frac{\alpha}{2}$ and $ \omega_2(x)= g(x) (|x|^2 +1)^\frac{\beta}{2}$,  where $ g(x) $ is positive except at say a point, smooth and where $ \lim_{|x| \rightarrow \infty} g(x) = C \in (0,\infty)$      For this class of weights we can essentially obtain optimal results 


\begin{thm} \label{alpha_beta}   Take $ \omega_1 $ and $ \omega_2$ as above 
\begin{enumerate}

\item If $ N+ \alpha - 2 <0$ then there is no stable sub-solution for $(G)$, $(L)$ (here we require it to be positive) and in the case of $(M)$ there is no positive  stable  super-solution  This case is the trivial case, see Remark \ref{triv}   \\



\textbf{Assumption:} For the remaining cases we assume that $ N + \alpha -2 > 0$ 

  \item If  $N+\alpha-2<4(\beta-\alpha+2)$ then there is no  stable sub-solution for $ (G)$ 

\item If $N+\alpha-2<\frac{ 2(\beta-\alpha+2)   }{p-1} \left( p+\sqrt{p(p-1)}  \right)$ then there is  no positive stable sub-solution of $(L)$ 

\item If $N+\alpha-2<\frac{2(\beta-\alpha+2)   }{p+1} \left( p+\sqrt{p(p+1)}  \right)$ then there is no positive stable super-solution of $(M)$ 

\item Further more 2,3,4 are optimal in the sense if $ N + \alpha -2 > 0$ and the remaining inequality is not satisfied (and in addition we assume we don't have equality in the inequality) then we can find a suitable function $ g(x)$ which satisfies the above properties and a stable sub/super-solution $u$ for the appropriate equation 

\end{enumerate}

\end{thm}

\begin{remark}  Many of the above results can be extended to the case of equality in either the $ N + \alpha - 2 \ge 0$ and also the other inequality which depends on the equation we are  examining   We omit the details because one cannot prove the results in a unified way 

\end{remark}






















In showing that an explicit solution is stable we will need the   weighted Hardy inequality given in \cite{craig} 
\begin{lemma} \label{Har}
Suppose $ E>0$ is a smooth function   Then one has
\[ (\tau-\frac{1}{2})^2 \int E^{2\tau-2} | \nabla E|^2 \phi^2 + (\frac{1}{2}-\tau) \int (-\Delta E) E^{2\tau-1} \phi^2 \le \int E^{2\tau} | \nabla \phi|^2,\] for all $ \phi \in C_c^\infty({\mathbb{R}}^N)$ and $ \tau \in {\mathbb{R}}$ 
\end{lemma}  By picking an appropriate function $E$ this gives,

\begin{cor} \label{Hardy}
For all $ \phi \in C_c^\infty$ and $ t , \alpha \in {\mathbb{R}}$  We have
  \begin{eqnarray*}
\int (1+|x|^2)^\frac{\alpha}{2} |\nabla\phi|^2 &\ge& (t+\frac{\alpha}{2})^2 \int |x|^2 (1+|x|^2)^{-2+\frac{\alpha}{2}}\phi^2\\
&&+(t+\frac{\alpha}{2})\int  (N-2(t+1)   \frac{|x|^2}{1+|x|^2}) (1+|x|^2)^{-1+\frac{\alpha} {2}} \phi^2 
\end{eqnarray*}
 \end{cor}























\section{Proof of main results}

\textbf{ Proof of Theorem \ref{main_non_exist} }   (1)  Suppose $ u$ is a stable sub-solution of $(G)$ with $ I_G,J_G \rightarrow 0$ as $ R \rightarrow \infty$ and  let $ 0 \le \phi \le 1$ denote a smooth compactly supported function   Put $ \psi:= e^{tu} \phi$ into (\ref{stable}), where $ 0 <t<2$, to arrive at
\begin{eqnarray*}
\int \omega_2 e^{(2t+1)u} \phi^2 &\le & t^2 \int \omega_1 e^{2tu} | \nabla u|^2 \phi^2 \\
&& +\int \omega_1 e^{2tu}|\nabla \phi|^2 + 2 t \int \omega_1 e^{2tu} \phi \nabla u \cdot \nabla \phi 
\end{eqnarray*}  Now multiply $(G)$ by $ e^{2tu} \phi^2$ and integrate by parts to arrive at
\[ 2t \int \omega_1 e^{2tu} | \nabla u|^2 \phi^2 \le \int \omega_2 e^{(2t+1) u} \phi^2 - 2 \int \omega_1 e^{2tu} \phi \nabla u \cdot \nabla \phi,\]
and now if one equates like terms they arrive at
\begin{eqnarray} \label{start}
\frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^2  & \le & \int \omega_1 e^{2tu} \left( | \nabla \phi |^2 - \frac{ \Delta \phi}{2} \right) dx \nonumber \\
&& - \frac{1}{2} \int e^{2tu} \phi \nabla \omega_1 \cdot \nabla \phi 
\end{eqnarray}   Now substitute $ \phi^m$ into this inequality for $ \phi$ where $ m $ is a big integer to obtain
\begin{eqnarray} \label{start_1}
\frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^{2m}  & \le & C_m \int \omega_1 e^{2tu} \phi^{2m-2}  \left( | \nabla \phi |^2 + \phi |\Delta \phi|  \right) dx \nonumber \\
&& - D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi
\end{eqnarray} where $ C_m$ and $ D_m$ are positive constants just depending on $m$    We now estimate the  terms on the right but we mention that when ones assume the appropriate monotonicity on $ \omega_1$ it is the last integral on the right which one is able to drop 

\begin{eqnarray*}
\int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi|^2 & = & \int \omega_2^\frac{2t}{2t+1} e^{2tu} \phi^{2m-2}  \frac{ \omega_1 }{\omega_2^\frac{2t}{2t+1}} | \nabla \phi|^2  \\
& \le &  \left( \int \omega_2 e^{(2t+1) u} \phi^{(2m-2) \frac{(2t+1)}{2t}} dx \right)^\frac{2t}{2t+1}\\ &&\left( \int \frac{ \omega_1 ^{2t+1}}{\omega_2^{2t}} | \nabla \phi |^{2(2t+1)} \right)^\frac{1}{2t+1} 
\end{eqnarray*}
Now, for fixed $ 0 <t<2$ we can take $ m $ big enough so $ (2m-2) \frac{(2t+1)}{2t} \ge 2m $ and since $ 0 \le \phi \le 1$ this allows us to replace the power on $ \phi$ in the first term on the right with $2m$   and hence we obtain
 \begin{equation} \label{three}
 \int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi|^2  \le \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{2t}{2t+1} \left( \int \frac{ \omega_1 ^{2t+1}}{\omega_2^{2t}} | \nabla \phi |^{2(2t+1)} \right)^\frac{1}{2t+1} 
 \end{equation}    We now take the test functions $ \phi$ to be such that $ 0 \le  \phi \le 1$ with $ \phi $ supported in the ball $ B_{2R}$ with $ \phi = 1 $ on $ B_R$ and $ | \nabla \phi | \le \frac{C}{R}$ where $ C>0$ is independent of $ R$    Putting this choice of $ \phi$ we obtain
 \begin{equation} \label{four}
 \int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi |^2 \le \left( \int \omega_2 e^{(2t+1)u} \phi^{2m} \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1} 
 \end{equation}  One similarly shows that
 \[ \int \omega_1 e^{2tu} \phi^{2m-1} | \Delta \phi| \le \left( \int \omega_2 e^{(2t+1)u} \phi^{2m} \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1} \]
 So, combining the results we obtain

 \begin{eqnarray} \label{last} \nonumber \frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^{2m} &\le& C_m \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}\\
 &&- D_m \int e^{2tu} \phi^{2m-1}  \nabla \omega_1 \cdot \nabla \phi 
 \end{eqnarray}
 We now estimate this last term   A similar argument using H\"{o}lder's inequality shows that
 \[ \int e^{2tu} \phi^{2m-1} | \nabla \omega_1| | \nabla \phi| \le \left(  \int \omega_2 \phi^{2m} e^{(2t+1) u} dx \right)^\frac{2t}{2t+1} J_G^\frac{1}{2t+1}  \] Combining the results gives that
\begin{equation} \label{last}
(2-t) \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{1}{2t+1} \le I_G^\frac{1}{2t+1} + J_G^\frac{1}{2t+1},
\end{equation} and now we send $ R \rightarrow \infty$ and use the fact that $ I_G, J_G \rightarrow 0$ as $ R \rightarrow \infty$ to see that
\[ \int \omega_2 e^{(2t+1) u} =0, \] which is clearly a contradiction    Hence there is no stable sub-solution of $(G)$ 

(2)    Suppose that $u >0$ is a stable sub-solution (super-solution) of $(L)$   Then a similar calculation as in (1) shows that for  $ p - \sqrt{p(p-1)} <t < p + \sqrt{p(p-1)}$,  $( 0 <t<\frac{1}{2})$ one has

\begin{eqnarray}   \label{shit}
(p  -\frac{t^2}{2t-1}   )\int \omega_2 u^{2t+p-1} \phi^{2m} & \le & D_m \int \omega_1 u^{2t} \phi^{2(m-1)} (|\nabla\phi|^2  +\phi |\Delta \phi |) \nonumber \\
&& +C_m \frac{(1-t)}{2(2t-1)} \int u^{2t} \phi^{2m-1}\nabla \omega_1 \cdot  \nabla \phi 
 \end{eqnarray}    One now applies H\"{o}lder's argument as in (1) but the terms $ I_L$ and $J_L$ will appear on the right hand side of the resulting
 equation       This shift from a sub-solution to a super-solution depending on whether $ t >\frac{1}{2}$ or $ t < \frac{1}{2}$ is a result from the sign change of $ 2t-1$ at $ t = \frac{1}{2}$    We leave the details for the reader 


(3)   This case is also similar to (1) and (2) 


\hfill $ \Box$

 \textbf{Proof of Theorem \ref{mono} }   (1)   Again we suppose there is a stable sub-solution $u$ of $(G)$   Our starting point  is (\ref{start_1}) and we wish to be able to drop the term
 \[ - D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi, \] from (\ref{start_1})     We can choose $ \phi$ as in the proof of Theorem \ref{main_non_exist} but also such that $ \nabla \phi(x) = - C(x) x$ where $ C(x) \ge 0$      So if we assume that $ \nabla \omega_1 \cdot x \le 0$ for big $x$ then we see that this last term is non-positive and hence we can drop the term   The the proof is as before but now we only require that $ \lim_{R \rightarrow \infty} I_G=0$ 

 (2)      Suppose that $ u >0$ is a stable sub-solution of $(L)$  and so (\ref{shit}) holds for all $  p - \sqrt{p(p-1)} <t< p + \sqrt{p(p-1)}$    Now we wish to use monotonicity to drop the term from (\ref{shit}) involving the term $ \nabla \omega_1 \cdot \nabla \phi$       $ \phi$ is chosen the same as in (1)  but here one notes that the co-efficient for this term changes sign at $ t=1$ and hence by restriction $t$ to the appropriate side of 1 (along with the above condition on $t$ and $\omega_1$) we can drop the last term depending on which monotonicity we have and hence to obtain a contraction we only require that $ \lim_{R \rightarrow \infty} I_L =0$    The result for the non-existence of a stable super-solution is similar be here one restricts $ 0 < t < \frac{1}{2}$ 


(3)   The proof here is similar to (1) and (2) and we omit the details 




 \hfill $\Box$













\textbf{Proof of Corollary \ref{thing} }   We suppose  that $ \omega_1 \le C  \omega_2$  for big $ x$, $ \omega_2 \in L^\infty$,  $ \nabla \omega_1(x) \cdot  x \le 0$ for big $ x$      \\
(1)     Since $ \nabla \omega_1 \cdot x \le 0$ for big $x$ we can apply Theorem \ref{mono} to show the non-existence of a stable solution to $(G)$    Note that  with the above assumptions on $ \omega_i$ we have that
\[ I_G \le \frac{C R^N}{R^{4t+2}} \]  For $ N \le 9$  we can take $ 0 <t<2$  but close enough to $2$ so the right hand side goes to zero as $ R \rightarrow \infty$ 

Both (2) and (3) also follow directly from applying Theorem \ref{mono}    Note that one can say more about (2) by taking the multiple cases as listed in Theorem \ref{mono} but we have choice to leave this to the reader 


\hfill $ \Box$


\textbf{Proof of Corollary \ref{po} }   Since we have no monotonicity conditions now we will need both $I$ and $J$ to go to zero to show the non-existence of a stable solution    Again the results are obtained immediately by applying Theorem \ref{main_non_exist}  and we prefer to omit the details 


\hfill $\Box$
















\textbf{Proof of Theorem \ref{alpha_beta} }  (1)   If $ N + \alpha -2 <0$ then using Remark \ref{triv}  one easily sees there is no stable sub-solution of $(G)$ and $(L)$ (positive for $(L)$) or a positive stable super-solution of $(M)$    So we now assume that $ N + \alpha -2 > 0$      Note that the monotonicity of $ \omega_1$ changes when $ \alpha $ changes sign and hence one would think that we need to consider separate cases if we hope to utilize the monotonicity results    But a computation shows that in fact $ I$ and $J$ are just multiples of each other in all three cases so it suffices to show, say, that $ \lim_{R \rightarrow \infty} I =0$  \\
(2)    Note that for $ R >1$ one has
\begin{eqnarray*}
I_G & \le  & \frac{C}{R^{4t+2}} \int_{R <|x| < 2R} |x|^{ \alpha (2t+1) - 2t \beta} \\
& \le  &  \frac{C}{R^{4t+2}}  R^{N + \alpha (2t+1) - 2t \beta},
\end{eqnarray*} and so to show the non-existence we want to find some $ 0 <t<2$ such that
$  4t+2 > N  + \alpha(2t+1) - 2 t \beta$,   which is equivalent to  $ 2t ( \beta - \alpha +2) > (N + \alpha -2)$     Now recall that we are assuming that $ 0 < N + \alpha -2 < 4 ( \beta - \alpha +2) $ and hence we have the desired result by taking $  t <2$ but sufficiently close 
The proof of the non-existence results for
(3) and (4) are similar and we omit the details    \\
(5)   We now assume that $N+\alpha-2>0$   In showing the existence of stable sub/super-solutions we need to consider  $ \beta - \alpha + 2 <0$ and $ \beta - \alpha +2 >0$ separately 

\begin{itemize} \item $(\beta - \alpha + 2 <0)$  Here we take $ u(x)=0$ in the case of $(G)$ and $ u=1$ in the case of $(L)$ and $(M)$  In addition we take $ g(x)=\E$   It is clear that in all cases $u$ is the appropriate sub or super-solution   The only thing one needs to check is the stability     In all cases this reduces to trying to show that we have
\[ \sigma \int       (1+|x|^2)^{\frac{\alpha}{2}  -1}    \phi^2 \le \int      (1+|x|^2)^{\frac{\alpha}{2}}   | \nabla\phi  |^2,\]  for all $ \phi \in C_c^\infty$ where  $ \sigma $ is some small positive constant; its either $ \E$ or $ p \E$ depending on which equation were are examining 
To show this we use the result from Corollary \ref{Hardy} and we drop a few positive terms to arrive at
\begin{equation*}
\int (1+|x|^2)^\frac{\alpha}{2} |\nabla\phi|^2\ge (t+\frac{\alpha}{2})\int \left (N-2(t+1)   \frac{|x|^2}{1+|x|^2}\right) (1+|x|^2)^{-1+\frac{\alpha} {2}}
\end{equation*} which holds for all $ \phi \in C_c^\infty$ and $ t,\alpha \in {\mathbb{R}}$ 
  Now, since $N+\alpha-2>0$, we can choose $t$ such that $-\frac{\alpha}{2}<t<\frac{n-2}{2}$   So, the integrand function in the right hand side is positive and since for small enough $\sigma$ we have
  \begin{equation*}
\sigma \le  (t+\frac{\alpha}{2})(N-2(t+1)   \frac{|x|^2}{1+|x|^2})  \ \ \ \text {for all} \ \ x\in \mathbb{R}^N
\end{equation*}
 we get stability 



\item ($\beta-\alpha+2>0$) In the case of $(G)$ we take   $u(x)=-\frac{\beta-\alpha+2}{2} \ln(1+|x|^2)$ and $g(x):= (\beta-\alpha+2)(N+(\alpha-2)\frac{|x|^2}{1+|x|^2})$  By a computation one sees that $u$ is a sub-solution of $(G)$ and hence we need now to only show the stability, which amounts to showing that
\begin{equation*}
\int \frac{g(x)\psi^2}{(1+|x|^{2   })^{-\frac{\alpha}{2}+1}}\le \int\frac{|\nabla\psi|^2}{    (1+|x|^2)^{-\frac{\alpha}{2}}     },
\end{equation*} for all $ \psi \in C_c^\infty$   To show this we use  Corollary \ref{Hardy}   So  we  need to choose an appropriate $t$ in   $-\frac{\alpha}{2}\le t\le\frac{N-2}{2}$  such that for all $x\in {\mathbb{R}}^N$ we have
 \begin{eqnarray*}
 (\beta-\alpha+2)\left(    N+  (\alpha-2)\frac{|x|^2}{1+|x|^2}\right)         &\le& (t+\frac{\alpha}{2})^2 \frac{ |x|^2 }{(1+|x|^2}\\
&&+(t+\frac{\alpha}{2}) \left(N-2(t+1)   \frac{|x|^2}{1+|x|^2}\right) 
\end{eqnarray*}
With a  simple calculation one sees we need just to have
   \begin{eqnarray*}
 (\beta-\alpha+2)&\le& (t+\frac{\alpha}{2}) \\
  (\beta-\alpha+2)     \left(    N+  \alpha-2\right)      &   \le&  (t+\frac{\alpha}{2}) \left(N-t-2+\frac{\alpha}{2}) \right) 
 \end{eqnarray*}     If one takes $ t= \frac{N-2}{2}$ in the case where $ N \neq 2$ and $ t $ close to zero in the case for $ N=2$ one easily sees the above inequalities both hold, after considering all the constraints on $ \alpha,\beta$ and $N$ 

 We now consider the case of $(L)$   Here one takes $g(x):=\frac {\beta-\alpha+2}{p-1}(    N+  (\alpha-2-\frac{\beta-\alpha+2}{p-1})
\frac{|x|^2}{1+|x|^2})$ and $ u(x)=(1+|x|^2)^{ -\frac  {\beta-\alpha+2}{2(p-1)} }$    Using essentially the same approach as in $(G)$ one shows that $u$ is a stable sub-solution of $(L)$ with this choice of $g$    \\
For the case of $(M)$ we take   $u(x)=(1+|x|^2)^{ \frac  {\beta-\alpha+2}{2(p+1)}   }$ and $g(x):=\frac {\beta-\alpha+2}{p+1}(    N+  (\alpha-2+\frac{\beta-\alpha+2}{p+1})
\frac{|x|^2}{1+|x|^2})$ 



\end{itemize}


\hfill  $  \Box$












ground_truth: $\int f'(u) \psi^2 \le \int | \nabla \psi|^2, \forall \psi \in C_c^2$.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the conduction gap depend on the strain direction?

Read the following text and answer briefly.

Context 2: \section{Introduction}

In spite of being an attractive material with excellent electronic properties \cite{ahcn09},  practical applications of graphene as in conventional semiconductor devices are still questionable due to its gapless nature  In particular, the ON/OFF current ratio is low while the saturation of current is poor in pristine graphene transistors \cite{schw10}  Many efforts of bandgap engineering in graphene \cite{yhan07,khar11,lher13,jbai10,zhan09} have been made to solve these issues  The pioneer technique proposed \cite{yhan07} is to cut 2D graphene sheets into 1D narrow nanoribons  In 2D graphene sheets, some options as Bernal-stacking of graphene on hexagonal boron nitride substrate \cite{khar11}, nitrogen-doped graphene \cite{lher13}, graphene nanomesh lattice \cite{jbai10,berr13} and Bernal-stacking bilayer graphene \cite{zhan09} have been explored  However, the possibility to open a sizable bandgap in graphene as large as those of standard semiconductors is still very unlikely  In particular, it requires a very good control of lattice geometry and edge disorder in narrow graphene nanoribbons (GNRs) \cite{quer08} and in graphene nanomesh lattices \cite{hung13}, while the bandgap opening in bilayer graphene by a perpendicular electric field may not be large enough for realistic applications \cite{fior09}  Other methods should be further verified by experiments 

\begin{figure}[!t]
\centering
\includegraphics[width=2 8in]{Fig01 pdf}
\caption{Schematic of unstrained/strained graphene junctions investigated in this work }
\label{fig_sim1}
\end{figure}

On the other hand, graphene was experimentally demonstrated to be able to sustain a much larger strain than conventional semiconductors, making it a promising candidate for flexible electronics (see in a recent review \cite{shar13})  Indeed, strain engineering has been suggested to be an alternative approach to modulating efficiently the electronic properties of graphene nanomaterials  In particular, the bandgap has periodic oscillations in the armchair GNRs \cite{ylu210} while the spin polarization at the ribbon edges (and also the bandgap) can be modulated by the strain in the zigzag cases  In 2D graphene sheets, a finite gap can open under large strains, otherwise, it may remain close to zero but the Dirac points are displaced \cite{cocc10,per209,pere09,huan10}  Many interesting electrical, optical, and magnetic properties induced by strain in graphene have been also explored, e g  see in \cite{bunc07,pere09,kuma12,per010,pell10,guin10,tlow10,zhai11} 

Besides, local strain is a good option to improve the electrical performance of graphene devices \cite{pere09,ylu010,fuji10,juan11,baha13}  For instance, it has been shown to enhance the ON current in a GNR tunneling FET \cite{ylu010} and to fortify the transport gap in GNR strained junctions \cite{baha13}  In a recent work \cite{hung14}, we have investigated the effects of uniaxial strain on the transport in 2D unstrained/strained graphene junctions and found that due to the strain-induced shift of Dirac points, a significant conduction gap of a few hundreds meV can open with a small strain of a few percent  This type of strained junction was then demonstrated to be an excellent candidate to improve the electronic operation of graphene transistors  It hence motivates us to further investigate the properties of this conduction gap so as to optimize the performance of graphene devices  On the one hand, the effects of strain should be, in principle, dependent on its applied direction  On the other hand, because the appearance of conduction gap is a consequence of the shift of Dirac points along the $k_y$-axis, it is predicted that this gap should also depend on the transport direction  Note that here, Oy (Ox) - axis is assumed to be perpendicular (parallel) to the transport direction  The effects of both strain and transport directions will be clarified systematically in the current work 

\section{Model and calculations}

In this work, the $\pi$-orbital tight binding model constructed in \cite{per209} is used to investigate the electronic transport through the graphene strained junctions schematized in Fig  1  The Hamiltonian is ${H_{tb}} = \sum\nolimits_{nm} {{t_{nm}}c_n^\dag {c_m}}$ where $t_{nm}$ is the hopping energy between nearest neighbor \emph{n}th and \emph{m}th atoms  The application of a uniaxial strain of angle $\theta$ causes the following changes in the $C-C$ bond vectors:
\begin{eqnarray}
 {{\vec r}_{nm}}\left( \sigma \right) &=& \left\{ {1 + {M_s}\left( \sigma, \theta \right)} \right\}{{\vec r}_{nm}}\left( 0 \right) \\
 {M_s}\left( \sigma, \theta \right) &=& \sigma \left[ {\begin{array}{*{20}{c}}
 {{{\cos }^2}\theta  - \gamma {{\sin }^2}\theta }&{\left( {1 + \gamma } \right)\sin \theta \cos \theta }\\
 {\left( {1 + \gamma } \right)\sin \theta \cos \theta }&{{{\sin }^2}\theta  - \gamma {{\cos }^2}\theta }
 \end{array}} \right] \nonumber
\end{eqnarray}
where $\sigma$ represents the strain and $\gamma \simeq 0 165$ is the Poisson ratio \cite{blak70}  The hopping parameters are defined as $t_{nm} \left( \sigma \right) = t_0 \exp\left[-3 37\left(r_{nm} \left( \sigma \right) /r_0 - 1\right)\right]$, where the hopping energy $t_0 = -2 7$ $eV$ and the bond length $r_{nm} \left( 0 \right) \equiv r_0 = 0 142$ $nm$ in the unstrained case  Therefore, there are three different hoping parameters $t_{1,2,3}$ corresponding to three bond vectors ${\vec r}_{1,2,3}$, respectively, in the strained graphene part of the structure (see Fig  1)  Here, we assume a 1D profile of applied strain, i e , the strain tensor is a function of position along the transport direction Ox while it is constant along the Oy-axis  The transport direction, $\phi$, and strain direction, $\theta$, are determined as schematized in Fig  1  Based on this tight binding model, two methods described below can be used to investigate the conduction gap of the considered strained junctions 

\textbf{Green's function calculations } First, we split the graphene sheet into the smallest possible unit cells periodically repeated along the Ox/Oy directions with the indices $p/q$, respectively (similarly, see the details in \cite{hung12})  The tight-binding Hamiltonian can therefore be expressed in the following form:
\begin{eqnarray}
{H_{tb}} = \sum\limits_{p,q} {\left( {{H_{p,q}} + \sum\limits_{{p_1},{q_1}} {{H_{p,q \to p_1,q_1}}} } \right)}
\end{eqnarray}
where $H_{p,q}$ is the Hamiltonian of cell $\{p,q\}$, and $H_{p,q \to p_1,q_1}$ denotes the coupling of cell $\{p,q\}$ to its nearest neighbor cell $\{p_1,q_1\}$  We then Fourier transform the operators in Eq  (2) as follows:
\begin{eqnarray}
 {c_{p,q}} = \frac{1}{{\sqrt {{M_{cell}}} }}\sum\limits_{{\kappa_y}} {{e^{i{q\kappa_y}}}} {{\hat c}_{p,{\kappa_y}}},
\end{eqnarray}
where $M_{cell}$ is the number of unit cells and $\kappa_y \equiv k_y L_y$ with the size $L_y$ of unit cells along the Oy direction  The Hamiltonian (2) is finally rewritten as a sum of $\kappa_y$-dependent 1D-components:
\begin{eqnarray}
{H_{tb}} &=& \sum\limits_{{\kappa_y}} {\hat H\left( {{\kappa_y}} \right)} \\
\hat H\left( {{\kappa_y}} \right) &=& \sum\limits_p {{{\hat H}_{p \to p - 1}}\left( {{\kappa_y}} \right) + {{\hat H}_p}\left( {{\kappa_y}} \right) + {{\hat H}_{p \to p + 1}}}\left( {{\kappa_y}} \right) \nonumber
\end{eqnarray}
With this Hamiltonian form, the Green's function formalism can be easily applied to compute transport quantities in the graphene strained junction with different transport directions  In particular, the conductance at zero temperature is determined as:
\begin{eqnarray}
 \mathcal{G} \left( \epsilon \right) = \frac{{e^2 W}}{{\pi h L_y}}\int\limits_{BZ} {d{\kappa_y} \mathcal{T}\left( {\epsilon, {\kappa_y}} \right)}
\end{eqnarray}
where $\mathcal{T}\left( {\epsilon,{\kappa_y}} \right)$ is the transmission probability computed from the Green's functions  The integration over $\kappa_y$ is performed in the whole first Brillouin zone  As in ref  \cite{hung13}, the gap of conductance (conduction gap) is then measured from the obtained data of conductance 

\textbf{Bandstructure analyses } To determine the conduction gap of strained junctions, we find that another simple way based on the analysis of graphene bandstructures could be efficiently used  It is described as follows  Since the conductance is computed from Eq  (5), the appearance of conduction gap is essentially governed by the gaps of transmission probability, which is determined from the energy gaps in the unstrained and strained graphene sections  These energy gaps can be defined directly from the graphene bandstructures  Therefore, our calculation has two steps, similar to that in \cite{hung14}  From the graphene bandstructures obtained using the tight-binding Hamiltonian above, we first look for the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ for a given $\kappa_y$ of two graphene sections  The maximum of these energy gaps determines the gap $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ of transmission probability through the junction  Finally, the conduction gap $E_{cond gap}$ is obtained by looking for the minimum value of $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ when varying $\kappa_y$ in the whole Brillouin zone 

In particular, the energy bands of strained graphene are given by
\begin{eqnarray}
 E\left( {\vec k} \right) =  \pm \left| {{t_1}{e^{i\vec k{{\vec a}_1}}} + {t_2}{e^{i\vec k{{\vec a}_2}}} + {t_3}} \right|
\end{eqnarray}
where the plus/minus sign corresponds to the conduction/valence bands, respectively  For a given direction $\phi$ of transport, in principle, the vectors $\vec L_{x,y}$ defining the sizes of unit cell along the Ox and Oy directions, respectively, can be always expressed as ${\vec L_x} = {n_1}{\vec a_1} + {n_2}{\vec a_2}$ and ${\vec L_y} = {m_1}{\vec a_1} + {m_2}{\vec a_2}$ with $\cos \phi = \frac{{{{\vec L}_x}\vec L_x^0}}{{{L_x}L_x^0}}$ and $\sin \phi = \frac{{{{\vec L}_x}\vec L_y^0}}{{{L_x}L_y^0}}$ while $\vec L_{x,y}^0 = {\vec a_1} \pm {\vec a_2}$  Note that $n_{1,2}$ and $m_{1,2}$ are integers while $\frac{{{m_1}}}{{{m_2}}} =  - \frac{{{n_1} + 2{n_2}}}{{{n_2} + 2{n_1}}}$, i e , ${\vec L_{x}} {\vec L_{y}} = 0$  In other words, we have the following expressions
\begin{eqnarray}
{{{\vec a}_1} = \frac{{ - {m_2}{{\vec L}_x} + {n_2}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}};\,\,\,{{\vec a}_2} = \frac{{{m_1}{{\vec L}_x} - {n_1}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}}}
\end{eqnarray}
On this basis, the energy bands can be rewritten in terms of $\kappa_{x, y} = \vec k \vec L_{x,y} \left( { \equiv {k_{x,y}}{L_{x,y}}} \right)$ by substituting Eqs  (7) into Eq  (6)  This new form of energy bands is finally used to compute the conduction gap of strained junctions 

As a simple example, in the case of $\phi = 0$ (armchair direction), we calculate the conduction gap as follows  First, Eq  (6) is rewritten in the form
\begin{eqnarray}
 E_{\phi = 0}\left( {\vec \kappa} \right) =  \pm \left| {{t_1}{e^{i\kappa_y/2}} + {t_2}{e^{ - i\kappa_y/2}} + {t_3}{e^{ - i\kappa_x/2}}} \right|
\end{eqnarray}
with the vectors $\vec L_{x,y} \equiv \vec L_{x,y}^0$  Using this new form, the energy gap of strained graphene for a given $\kappa_y$ is determined as
\begin{equation}
{E_{strain}^{gap}}\left( {{\kappa_y}} \right) = 2 \left| {\sqrt {{{\left( {{t_1} - {t_2}} \right)}^2} + 4{t_1}{t_2}{{\cos }^2}\frac{{{\kappa_y}}}{2}}  + {t_3}} \right|
\end{equation}
while ${E_{unstrain}^{gap}}\left( {{\kappa_y}} \right)$ is given by the same formula with $t_1$ = $t_2$ = $t_3$ $\equiv$ $t_0$  The gap of transmission probability through the junction is then determined as ${E_{junc}^{gap}}\left( {{\kappa_y}} \right) = \max \left[ {E_{unstrain}^{gap}\left( {{\kappa_y}} \right),E_{strain}^{gap}\left( {{\kappa_y}} \right)} \right]$ and, finally, the conduction gap is given by ${E_{cond gap}} = \min \left[ {E_{junc}^{gap}\left( {{\kappa_y}} \right)} \right]$ for $\kappa_y$ in the whole Brillouin zone 

We would like to notice that the Green's function calculations and the banstructure analyses give the same results of conduction gap in the junctions where the transition region between unstrained and strained graphene sections is long enough, i e , larger than about 5 to 6 nm  In the case of short length, as discussed in \cite{baha13,hung14}, this transition zone can have significant effects on the transmission between propagating states beyond the energy gaps and hence can slightly enlarge the gap of conductance, compared to the results obtained from the bandstructure calculations 

\section{Results and discussion}

\begin{figure}[!t]
\centering
\includegraphics[width=3 0in]{Fig02 pdf}
\caption{Dependence of graphene bandgap (in the unit of eV) on the applied strain and its direction: tensile (a) and compressive (b)  The radius from the central point indicates the strain strength ranging from 0 (center) to 30 $\%$ (edge of maps) while the graphene lattice is superimposed to show visibly the strain direction  The orange circle corresponds to the strains of $\sigma = 23 \%$ }
\label{fig_sim2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3 4in]{Fig03 pdf}
\caption{Conductance ($G_0 = e^2W/hL_y$) as a function of energy in graphene strained junctions for $\sigma = 4 \%$ with different strain directions  The transport along the armchair direction ($\phi = 0$) is considered  The data obtained in a uniformly strained graphene is displayed for the comparison }
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 8in]{Fig04 pdf}
\caption{Local density of states (left panels) and corresponding transmission coefficient (right panels) for three different wave-vectors $k_y$ obtained in an unstrained/strained graphene junction of $\sigma = 4 \%$, and $\theta \equiv \phi = 0$  On the top is a schematic of graphene bandedges illustrating the strain-induced shift of Dirac points along the $k_y$-direction }
\label{fig_sim4}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 6in]{Fig05 pdf}
\caption{Maps of conduction gap in unstrained/strained graphene junctions: tensile (a,c) and compressive cases (b,d)  The transport is along the armchair $\phi = 0$ (a,b) and zigzag $\phi = 30^\circ$ directions (c,d)  The strain strength ranges from 0 (center) to 6 $\%$ (edge of maps) in all cases }
\label{fig_sim4}
\end{figure*}
First, we re-examine the formation of the bandgap of graphene under a uniaxial strain  From Eq  (9), it is shown that a strain-induced finite-bandgap appears only if ${E_{strain}^{gap}}\left( {{\kappa_y}} \right) > 0$ for all $k_y$ in the first Brillouin zone, i e , ${k _y} \in \left[ { - \frac{\pi}{L_y}, \frac{\pi}{L_y}} \right]$, otherwise, the bandgap remains zero  Hence, the condition for the bandgap to be finite is either
\begin{equation*}
 \left| {{t_1} - {t_2}} \right| > \left| {{t_3}} \right|\,\,\,\,\,{\rm{OR}}\,\,\,\,\,\left| {{t_3}} \right| > \left| {{t_1} + {t_2}} \right|
\end{equation*}
and the corresponding values of bandgap are
\begin{equation*}
 {E_{gap}} = 2\left( {\left| {{t_1} - {t_2}} \right| - \left| {{t_3}} \right|} \right)\,\,\,\,\,{\rm{OR}}\,\,\,\,\,2\left( {\left| {{t_3}} \right| - \left| {{t_1} + {t_2}} \right|} \right)
\end{equation*}
This result was actually reported in \cite{per209,hase06}  We remind as displayed in Fig  2(a) that a finite bandgap opens only for strain larger than $\sim 23 \%$ and the zigzag (not armchair) is the preferred direction for bandgap opening under a tensile strain \cite{per209}  We extend our investigation to the case of compressive strain and find  (see in Fig  2(b)) that (i) the same gap threshold of $\sigma \simeq 23 \%$ is observed but (ii) the preferred direction to open the gap under a compressive strain is the armchair, not the zigzag as the case of tensile strain  This implies that the properties of graphene bandstructure at low energy should be qualitatively the same when applying strains of $\left\{ {\sigma ,\theta } \right\}$ and of $\left\{ {-\sigma ,\theta + 90^\circ} \right\}$  This feature can be understood by considering, for example, strains of $\left\{ {\sigma , \theta = 0} \right\}$ and of $\left\{ {-\sigma , \theta = 90^\circ} \right\}$  Indeed, these strains result in the same qualitative changes on the bond-lengths, i e , an increased bond-length $r_3$ and reduced bond-lengths $r_{1,2}$  However, for the same strain strength, because of the exponential dependence of hoping energies on the bond-lengths, the compressive strain generally induces a larger bandgap than the tensile one, as can be seen when comparing the data displayed in Figs  2(a) and 2(b)  To conclude, we would like to emphasize that a large strain is necessary to open a bandgap in graphene  This could be an issue for practical applications, compared to the use of graphene strained junctions explored in \cite{hung14} 

We now go to explore the properties of conduction gap in the graphene strained junctions  In Fig  3, we display the conductance as a function of energy computed from Eq  (5) using the Green's function technique  As discussed above, a small strain of a few percent (e g , 4 $\%$ here) can not change the gapless character of graphene, i e , there is no gap of conductance in the case of uniformly strained graphene  However, similar to that reported in \cite{hung14}, a significant conduction-gap of a few hundreds meV can open in the unstrained/strained graphene junctions  The appearance of this conduction gap, as mentioned previously, is due to the strain-induced shift of Dirac points and is explained as follows  Actually, the strain causes the lattice deformation and can result in the deformation of graphene bandstructure  Therefore, the bandedges as a function of wave-vector $k_y$ in unstrained and strained graphene can be illustrated schematically as in the top panel of Fig  4  As one can see, the shift of Dirac points leads to the situation where there is no value of $\kappa_y$, for which the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are simultaneously equal to zero  This means that the transmission probability always shows a finite gap for any $\kappa_y$  For instance, the energy gap is zero (or small) in the unstrained (resp  strained) graphene section but finite in the strained (resp  unstrained) one in the vicinity of Dirac point $k_y = K_{unstrain}$ (resp  $K_{strain}$)  Accordingly, as illustrated in the pictures of LDOS in the left panels of Fig  4 and confirmed in the corresponding transmissions in the right panels, clear gaps of transmission are still obtained  Far from these values of $k_y$, $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are both finite (e g , see the LDOS plotted for $k_y = K_{gap}$) and hence a finite gap of transmission also occurs  On this basis, a finite gap of conductance is achieved  More important, Fig  3 shows that besides the strength of strain, the strain effect is also strongly dependent on the applied direction  For instance, the conduction gap takes the values of $\sim$ 295, 172 and 323 meV for $\theta = 0$, $30^\circ$ and $90^\circ$, respectively 

Below, we will discuss the properties of the conduction gap with respect to the strain, its applied direction, and the direction of transport  Note that due to the lattice symmetry, the transport directions $\phi$ and $\phi + 60^\circ$ are equivalent while the applied strain of angle $\theta$ is identical to that of $\theta + 180^\circ$  Hence, the data obtained for $\phi$ ranging from $-30^\circ$ to $30^\circ$ and $\theta  \in \left[ {0^\circ ,180^\circ } \right]$ covers the properties of conduction gap in all possible cases 

In Fig  5, we present the maps of conduction gap with respect to the strain and its applied direction in two particular cases: the transport is either along the armchair ($\phi = 0$) or the zigzag ($\phi = 30^\circ$) directions  Both tensile and compressive strains are considered  Let us first discuss the results obtained in the armchair case  Figs  5(a,b) show that (i) a large conduction gap up to about 500 meV can open with a strain of 6 $\%$ and (ii) again the conduction gap is strongly $\theta$-dependent, in particular, its peaks occur at $\theta = 0$ or $90^\circ$ while the gap is zero at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain  In principle, the conduction gap is larger if the shift of Dirac points in the $\kappa_y$-axis is larger, as discussed above about Figs  3-4  We notice that the strain-induced shifts can be different for the six Dirac points of graphene \cite{kitt12} and the gap is zero when there is any Dirac point observed at the same $\kappa_y$ in the two graphene sections  From Eq  (9), we find that the Dirac points are determined by the following equations:
\begin{eqnarray*}
  {\cos}\frac{\kappa_y}{2} &=& \pm \frac{1}{2}\sqrt{\frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{{t_1}{t_2}}}}, \\
  \cos \frac{{\kappa_x}}{2} &=& \frac{{{t_1} + {t_2}}}{{\left| {{t_3}} \right|}}\cos \frac{{\kappa_y}}{2},\,\,\,\sin \frac{{\kappa_x}}{2} = \frac{{{t_2} - {t_1}}}{{\left| {{t_3}} \right|}}\sin \frac{{\kappa_y}}{2},
\end{eqnarray*}
which simplify into ${\cos}\frac{\kappa_y}{2} = \pm \frac{1}{2}$ and, respectively, $\cos \left( {\frac{{{\kappa _x}}}{2}} \right) = \mp 1$ in the unstrained case  Hence, the zero conduction gap is obtained if
\begin{equation*}
  \frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{4{t_1}{t_2}}} = \frac{1}{4}
\end{equation*}
Additionally, it is observed that the effects of a strain $\{\sigma,\theta\}$ are qualitatively similar to those of a strain $\{-\sigma,\theta+90^\circ\}$, i e , the peaks and zero values of conduction gap are obtained at the same $\theta$ in these two situations  To understand this, we analyze the strain matrix $M_s \left(\sigma,\theta\right)$ and find that in the case of small strains studied here, there is an approximate relationship between the bond lengths under these two strains, given by \[{r \left( \sigma, \theta \right)} - {r \left( -\sigma, \theta + 90^\circ\right)} \simeq \sigma \left( {1 - \gamma } \right) r_0,\] which is $\theta$-independent for all \emph{C-C} bond vectors  It implies that there is a fixed ratio between the hopping energies $t_i \left( \sigma, \theta \right)$ and $t_i \left( -\sigma, \theta + 90^\circ\right)$ and hence there is the similar shift of Dirac points in these two cases 
\begin{figure}[!t]
\centering
\includegraphics[width=3 4in]{Fig06 pdf}
\caption{Map showing the dependence of conduction gap on the directions ($\theta,\phi$) for $\sigma = 4 \%$  The top is a diagram illustrating the rotation of Dirac points in the \emph{k}-space with the change in the transport direction $\phi$ }
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 5in]{Fig07 pdf}
\caption{Maps of conduction gap obtained in tensile/compressive strained junctions  The transport along the armchair/zigzag directions is considered in (a,b)/(c,d), respectively  The strains $\sigma_c = -2 \%$ and $\sigma_t = 2 \%$ are applied in (a,c) while $\sigma_c = -1 \%$ and $\sigma_t = 3 \%$ in (b,d) }
\label{fig_sim4}
\end{figure*}

We now go to analyze the properties of conduction gap shown in Figs  5(c,d) where the transport is along the zigzag direction $\phi = 30^\circ$  In fact, the conduction gap in this case can reach a value as high as that of the case of $\phi = 0$ but has different $\theta$-dependence  In particular, the conduction gap has peaks at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain, where it is zero in the case of $\phi = 0$  It is also equal to zero at $\theta = 0$ and $\theta = 90^\circ$ where the peaks of conduction gap occur in the latter case of $\phi = 0$  The relationship between these two transport directions can be explained as follows  On the one hand, based on the analyses above for $\phi = 0$, we find that for a given strength of strain, a maximum shift of Dirac points along the $k_y$-axis corresponds to a minimum along the $k_x$-one and vice versa when varying the strain direction $\theta$  On the other hand, as schematized in the top of Fig  6 below, the change in the transport direction results in the rotation of the first Brillouin zone, i e , the $k_x$ (resp  $k_y$) axis in the case of $\phi = 30^\circ$ is identical to the $k_y$ (resp  $k_x$) axis in the case of $\phi = 0$  These two features explain essentially the opposite $\theta$-dependence of conduction gap for $\phi = 30^\circ$, compared to the case of $\phi = 0$ as mentioned  Again, we found the same qualitative behavior of conduction gap when applying the strains of $\{\sigma,\theta\}$ and $\{-\sigma,\theta+90^\circ\}$ 

Next, we investigate the conduction gap with respect to different transport directions $\phi$  We display a ($\theta,\phi$)-map of conduction gap for $\sigma = 4 \%$ in Fig  6 and, in the top, an additional diagram illustrating the rotation of Dirac points in the $k-$space with the change in the transport direction  It is clearly shown that (i) a similar scale of conduction gap is obtained for all different transport directions, (ii) there is a smooth and continuous shift of $E_{cond gap}-\theta$ behavior when varying $\phi$, and (iii) the same behavior of $E_{cond gap}$ is also observed when comparing the two transport directions of $\phi$ and $\phi+30^\circ$, similarly to the comparison above between $\phi = 0^\circ$ and $30^\circ$  The data plotted in Fig  6 additionally shows that $E_{cond gap}$ takes the same value in both cases of $\{\phi,\theta\}$ and $\{-\phi,-\theta\}$ with a remark that the strains of $-\theta$ and $180^\circ-\theta$ are identical  Moreover, the values of $\theta$ and $\phi$, for which the conduction gap has a peak or is equal to zero, have an almost linear relationship  In particular, the relationship for conduction gap peaks is approximately given by $\theta = \theta_A - \eta_s \phi$  For tensile strains, $\eta_s$ takes the values of $\sim 1 5667$ and $1 4333$ for $\theta_A = 0$ and $90^\circ$, respectively  On the opposite, it is about $1 4333$ and $1 5667$ for $\theta_A = 0$ and $90^\circ$, respectively, for compressive strain cases  All these features are consequences of the rotation of Dirac points in the $k$-space with respect to the transport direction $\phi$ as illustrated in the diagram on the top and the lattice symmetry of graphene 

Finally, we investigate other junctions based on compressive and tensile strained graphene sections  The idea is that in this type of strained junction, the shifts of Dirac points are different in two graphene sections of different strains, which offers the possibilities to use smaller strains to achieve a similar conduction gap, compared to the case of unstrained/strained junction  In Fig  7, we display the maps of conduction gap with respect to the directions of compressive ($\theta_c$) and tensile ($\theta_t$) strains in two cases of transport direction $\phi = 0$ (armchair) and $30^\circ$ (zigzag) for given strain strengths  Indeed, as seen in Fig  7(a,b), with smaller strains $\left\{ {{\sigma _c},{\sigma _t}} \right\} = \left\{ { - 2\% ,2\% } \right\}$ or $\left\{ { - 1\% ,3\% } \right\}$, similar conduction gap of about 310 meV can be achieved (see Figs  7(a,b)) while it requires a strain of 4 $\%$ in the unstrained/strained junctions discussed above  However, since the shift of Dirac points is strongly dependent on the direction of applied strains and the transport direction, the properties of conduction gap are more complicated than in the latter case  In particular, our calculations show that the preferred transport directions to achieve a large conduction gap are close to the armchair one  Otherwise, the conduction gap is generally smaller, similarly to the data for $\phi = 30^\circ$ compared to $\phi = 0$, as shown in Fig  7  Additionally, it is shown that the preferred directions of applied strains in the case of $\phi = 0$ are close to ${\theta _c} \equiv {\theta _t} = 0$ or $90^\circ$ 

\section{Conclusion}

Based on the tight binding calculations, we have investigated the effects of uniaxial strain on the transport properties of graphene strained junctions and discuss systematically the possibilities of achieving a large conduction gap with respect to the strain, its applied direction and the transport direction  It has been shown that due to the strain-induced deformation of graphene lattice and hence of graphene bandstructure, a finite conduction gap higher than 500 meV can be achieved for a strain of only 6 $\%$  Moreover, as a consequence of the shift of Dirac points along the $k_y$-axis, the conduction gap is strongly dependent not only on the strain strength but also on the direction of applied strain and the transport direction  A full picture of these properties of conduction gap has been presented and explained  The study hence could be a good guide for the use of this type of unstrained/strained graphene junction in electronic applications 

\textbf{\textit{Acknowledgment }} This research in Hanoi is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103 02-1012 42  We also acknowledges the French ANR for financial support under the projects NANOSIM-GRAPHENE (Grant no  ANR-09-NANO-016) and MIGRAQUEL (Grant no  ANR-10-BLAN-0304) 





ground_truth: Peaks occur at certain strain directions, while the gap is zero at others.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are the datasets used in this community for research?

Read the following text and answer briefly.

Context 3: \section{Introduction}
Underwater robot picking is to use the robot to automatically capture sea creatures like holothurian, echinus, scallop, or starfish in an open-sea farm where underwater object detection is the key technology for locating creatures  Until now, the datasets used in this community are released by the Underwater Robot Professional Contest (URPC$\protect\footnote{Underwater Robot Professional Contest: {\bf http://en cnurpc org} }$) beginning from 2017, in which URPC2017 and URPC2018 are most often used for research  Unfortunately, as the information listed in Table \ref{Info}, URPC series datasets do not provide the annotation file of the test set and cannot be downloaded after the contest  
Therefore, researchers \cite{2020arXiv200511552C,2019arXiv191103029L} first have to divide the training data into two subsets, including a new subset of training data and a new subset of testing data, and then train their proposed method and other \emph{SOTA} methods  On the one hand, training other methods results in a significant increase in workload  On the other hand, different researchers divide different datasets in different ways, 
\begin{table}[t]
\renewcommand\tabcolsep{3 5pt}
\caption{Information about all the collected datasets  * denotes the test set's annotations are not available  \emph{3} in Class means three types of creatures are labeled, \emph{i e ,} holothurian, echinus, and scallop  \emph{4} means four types of creatures are labeled (starfish added)  Retention represents the proportion of images that retain after similar images have been removed }
\centering 
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Dataset&Train&Test&Class&Retention&Year \\ 
\hline 
URPC2017&17,655&985*&3&15\%&2017 \\
\hline
URPC2018&2,901&800*&4&99\%&2018 \\
\hline
URPC2019&4,757&1,029*&4&86\%&2019 \\
\hline
URPC2020$_{ZJ}$&5,543&2,000*&4&82\%&2020 \\
\hline
URPC2020$_{DL}$&6,575&2,400*&4&80\%&2020 \\
\hline
UDD&1,827&400&3&84\%&2020 \\
\hline 

\end{tabular}
\label{Info}
\end{table}
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=1\linewidth]{example pdf}
\end{center}
   \caption{Examples in DUO, which show a variety of scenarios in underwater environments }
\label{exam}
\end{figure*}
causing there is no unified benchmark to compare the performance of different algorithms 
In terms of the content of the dataset images, there are a large number of similar or duplicate images in the URPC datasets  URPC2017 only retains 15\% images after removing similar images compared to other datasets  Thus the detector trained on URPC2017 is easy to overfit and cannot reflect the real performance 
For other URPC datasets, the latter also includes images from the former, \emph{e g }, URPC2019 adds 2,000 new images compared to URPC2018; compared with URPC2019, URPC2020$_{ZJ}$ adds 800 new images  The URPC2020$_{DL}$ adds 1,000 new images compared to the URPC2020$_{ZJ}$  It is worth mentioning that the annotation of all datasets is incomplete; some datasets lack the starfish labels and it is easy to find error or missing labels  \cite{DBLP:conf/iclr/ZhangBHRV17} pointed out that although the CNN model has a strong fitting ability for any dataset, the existence of dirty data will significantly weaken its robustness 
Therefore, a reasonable dataset (containing a small number of similar images as well as an accurate annotation) and a corresponding recognized benchmark are urgently needed to promote community development 


To address these issues, we introduce a dataset called Detecting Underwater Objects (DUO) by collecting and re-annotating all the available underwater datasets  It contains 7,782 underwater images after deleting overly similar images and has a more accurate annotation with four types of classes (\emph{i e ,} holothurian, echinus, scallop, and starfish)  
Besides, based on the MMDetection$\protect\footnote{MMDetection is an open source object detection toolbox based on PyTorch  {\bf https://github com/open-mmlab/mmdetection}}$ \cite{chen2019mmdetection} framework, we also provide a \emph{SOTA} detector benchmark containing efficiency and accuracy indicators, providing a reference for both academic research and industrial applications  It is worth noting that JETSON AGX XAVIER$\protect\footnote{JETSON AGX XAVIER is an embedded development board produced by NVIDIA which could be deployed in an underwater robot  Please refer {\bf https://developer nvidia com/embedded/jetson-agx-xavier-developer-kit} for more information }$ was used to assess all the detectors in the efficiency test in order to simulate robot-embedded environment  DUO will be released in https://github com/chongweiliu soon 

In summary, the contributions of this paper can be listed as follows 

  $\bullet$ By collecting and re-annotating all relevant datasets, we introduce a dataset called DUO with more reasonable annotations as well as a variety of underwater scenes 

  $\bullet$ We provide a corresponding benchmark of \emph{SOTA} detectors on DUO including efficiency and accuracy indicators which could be a reference for both academic research and industrial applications  


\pagestyle{empty}
\section{Background}
In the year of 2017, underwater object detection for open-sea farming is first proposed in the target recognition track of Underwater Robot Picking Contest 2017$\protect\footnote{From 2020, the name has been changed into Underwater Robot Professional Contest which is also short for URPC }$ (URPC2017) which aims to promote the development of theory, technology, and industry of the underwater agile robot and fill the blank of the grabbing task of the underwater agile robot  The competition sets up a target recognition track, a fixed-point grasping track, and an autonomous grasping track  The target recognition track concentrates on finding the {\bf high accuracy and efficiency} algorithm which could be used in an underwater robot for automatically grasping 

The datasets we used to generate the DUO are listed below  The detailed information has been shown in Table \ref{Info} 

  {\bf URPC2017}: It contains 17,655 images for training and 985 images for testing and the resolution of all the images is 720$\times$405  All the images are taken from 6 videos at an interval of 10 frames  However, all the videos were filmed in an artificial simulated environment and pictures from the same video look almost identical  
  
   {\bf URPC2018}: It contains 2,901 images for training and 800 images for testing and the resolutions of the images are 586$\times$480, 704$\times$576, 720$\times$405, and 1,920$\times$1,080  The test set's annotations are not available  Besides, some images were also collected from an artificial underwater environment 
  
  {\bf URPC2019}: It contains 4,757 images for training and 1029 images for testing and the highest resolution of the images is 3,840$\times$2,160 captured by a GOPro camera  The test set's annotations are also not available and it contains images from the former contests 
  
  {\bf URPC2020$_{ZJ}$}: From 2020, the URPC will be held twice a year  It was held first in Zhanjiang, China, in April and then in Dalian, China, in August  URPC2020$_{ZJ}$ means the dataset released in the first URPC2020 and URPC2020$_{DL}$ means the dataset released in the second URPC2020  This dataset contains 5,543 images for training and 2,000 images for testing and the highest resolution of the images is 3,840$\times$2,160  The test set's annotations are also not available 
  
  {\bf URPC2020$_{DL}$}: This dataset contains 6,575 images for training and 2,400 images for testing and the highest resolution of the images is 3,840$\times$2,160  The test set's annotations are also not available 
  
  {\bf UDD \cite{2020arXiv200301446W}}: This dataset contains 1,827 images for training and 400 images for testing and the highest resolution of the images is 3,840$\times$2,160  All the images are captured by a diver and a robot in a real open-sea farm 

\begin{figure}[t]
\begin{center}
\includegraphics[width=1\linewidth]{pie pdf}
\end{center}
   \caption{The proportion distribution of the objects in DUO }
\label{pie}
\end{figure}



\begin{figure*}
  \centering
  \subfigure[]{\includegraphics[width=3 45in]{imagesize pdf}}
  \subfigure[]{\includegraphics[width=3 45in]{numInstance pdf}}
  \caption{(a) The distribution of instance sizes for DUO; (b) The number of categories per image }
  \label{sum}
\end{figure*}
\section{Proposed Dataset}

\subsection{Image Deduplicating}
As we explained in Section 1, there are a large number of similar or repeated images in the series of URPC datasets  Therefore, it is important to delete duplicate or overly similar images and keep a variety of underwater scenarios when we merge these datasets together  Here we employ the Perceptual Hash algorithm (PHash) to remove those images  PHash has the special property that the hash value is dependent on the image content, and it remains approximately the same if the content is not significantly modified  Thus we can easily distinguish different scenarios and delete duplicate images within one scenario  

After deduplicating, we obtain 7,782 images (6,671 images for training; 1,111 for testing)  The retention rate of the new dataset is 95\%, which means that there are only a few similar images in the new dataset  Figure \ref{exam} shows that our dataset also retains various underwater scenes 

\subsection{Image Re-annotation}
Due to the small size of objects and the blur underwater environment, there are always missing or wrong labels in the existing annotation files  In addition, some test sets' annotation files are not available and some datasets do not have the starfish annotation  In order to address these issues, we follow the next process which combines a CNN model and manual annotation to re-annotate these images  Specifically, we first train a detector (\emph{i e ,} GFL \cite{li2020generalized}) with the originally labeled images  After that, the trained detector predicts all the 7,782 images  We treat the prediction as the groundtruth and use it to train the GFL again  We get the final GFL prediction called {\bf the coarse annotation}  Next, we use manual correction to get the final annotation called {\bf the fine annotation}  Notably, we adopt the COCO \cite{Belongie2014} annotation form as the final format 
\subsection{Dataset Statistics}
{\bf The proportion of classes}: The total number of objects is 74,515  Holothurian, echinus, scallop, and starfish are 7,887, 50,156, 1,924, and 14,548, respectively  Figure \ref{pie} shows the proportion of each creatures where echinus accounts for 67 3\% of the total  The whole data distribution shows an obvious long-tail distribution because the different economic benefits of different seafoods determine the different breed quantities 

{\bf The distribution of instance sizes}: Figure \ref{sum}(a) shows an instance size distribution of DUO  \emph{Percent of image size} represents the ratio of object area to image area, and \emph{Percent of instance} represents the ratio of the corresponding number of objects to the total number of objects  Because of these small creatures and high-resolution images, the vast majority of objects occupy 0 3\% to 1 5\% of the image area 

{\bf The instance number per image}: Figure \ref{sum}(b) illustrates the number of categories per image for DUO  \emph{Number of instances} represents the number of objects one image has, and \emph{ Percentage of images} represents the ratio of the corresponding number of images to the total number of images  Most images contain between 5 and 15 instances, with an average of 9 57 instances per image 

{\bf Summary}:
In general, smaller objects are harder to detect  For PASCAL VOC \cite{Everingham2007The} or COCO \cite{Belongie2014}, roughly 50\% of all objects occupy no more than 10\% of the image itself, and others evenly occupy from 10\% to 100\%  
In the aspect of instances number per image, COCO contains 7 7 instances per image and VOC contains 3  In comparison, DUO has 9 57 instances per image and most instances less than 1 5\% of the image size 
Therefore, DUO contains almost exclusively massive small instances and has the long-tail distribution at the same time, which means it is promising to design a detector to deal with massive small objects and stay high efficiency at the same time for underwater robot picking 

\section{Benchmark}
Because the aim of underwater object detection for robot picking is to find {\bf the high accuracy and efficiency} algorithm, we consider both the accuracy and efficiency evaluations in the benchmark as shown in Table \ref{ben} 

\subsection{Evaluation Metrics}
Here we adopt the standard COCO metrics (mean average precision, \emph{i e ,} mAP) for the accuracy evaluation and also provide the mAP of each class due to the long-tail distribution 

{\bf AP} -- mAP at IoU=0 50:0 05:0 95 

{\bf AP$_{50}$} --  mAP at IoU=0 50 

{\bf AP$_{75}$} --  mAP at IoU=0 75  

{\bf AP$_{S}$} --   {\bf AP} for small objects of area smaller than 32$^{2}$ 

{\bf AP$_{M}$} --   {\bf AP} for objects of area between 32$^{2}$ and 96$^{2}$ 

{\bf AP$_{L}$} --   {\bf AP} for large objects of area bigger than 96$^{2}$ 

{\bf AP$_{Ho}$} --  {\bf AP} in holothurian 

{\bf AP$_{Ec}$} --  {\bf AP} in echinus 

{\bf AP$_{Sc}$} --  {\bf AP} in scallop 

{\bf AP$_{St}$} --  {\bf AP} in starfish 


For the efficiency evaluation, we provide three metrics:

{\bf Param } --  The parameters of a detector 

{\bf FLOPs} --  Floating-point operations per second 

{\bf FPS} --  Frames per second 

Notably, {\bf FLOPs} is calculated under the 512$\times$512 input image size and {\bf FPS} is tested on a JETSON AGX XAVIER under MODE$\_$30W$\_$ALL  

\subsection{Standard Training Configuration}
We follow a widely used open-source toolbox, \emph{i e ,} MMDetection (V2 5 0) to produce up our benchmark  During the training, the standard configurations are as follows:

  $\bullet$ We initialize the backbone models (\emph{e g ,} ResNet50) with pre-trained parameters on ImageNet \cite{Deng2009ImageNet} 

  $\bullet$ We resize each image into 512 $\times$ 512 pixels both in training and testing  Each image is flipped horizontally with 0 5 probability during training 

  $\bullet$ We normalize RGB channels by subtracting 123 675, 116 28, 103 53 and dividing by 58 395, 57 12, 57 375, respectively 

  $\bullet$ SGD method is adopted to optimize the model  The initial learning rate is set to be 0 005 in a single GTX 1080Ti with batchsize 4 and is decreased by 0 1 at the 8th and 11th epoch, respectively  WarmUp \cite{2019arXiv190307071L} is also employed in the first 500 iterations  Totally there are 12 training epochs 

  $\bullet$ Testing time augmentation (\emph{i e ,} flipping test or multi-scale testing) is not employed 



\subsection{Benchmark Analysis}
Table \ref{ben} shows the benchmark for the \emph{SOTA} methods  Multi- and one- stage detectors with three kinds of backbones (\emph{i e ,} ResNet18, 50, 101) give a comprehensive assessment on DUO  We also deploy all the methods to AGX to assess efficiency 

In general, the multi-stage (Cascade R-CNN) detectors have high accuracy and low efficiency, while the one-stage (RetinaNet) detectors have low accuracy and high efficiency  However, due to recent studies \cite{zhang2019bridging} on the allocation of more reasonable positive and negative samples in training, one-stage detectors (ATSS or GFL) can achieve both high accuracy and high efficiency 

\begin{table*}[htbp]
\renewcommand\tabcolsep{3 0pt}

\begin{center}
\caption{Benchmark of \emph{SOTA} detectors (single-model and single-scale results) on DUO  FPS is measured on the same machine with a JETSON AGX XAVIER under the same MMDetection framework, using a batch size of 1 whenever possible  R: ResNet } 
\label{ben}
\begin{tabular}{|l|l|c|c|c|ccc|ccc|cccc|}
\hline
Method&Backbone&Param &FLOPs&FPS&AP&AP$_{50}$&AP$_{75}$&AP$_{S}$&AP$_{M}$&AP$_{L}$&AP$_{Ho}$&AP$_{Ec}$&AP$_{Sc}$&AP$_{St}$ \\ 
\hline 
\emph{multi-stage:} &&&&&&&&&&&&&& \\

\multirow{3}{*}{Faster R-CNN \cite{Ren2015Faster}}
&R-18&28 14M&49 75G&5 7&50 1&72 6&57 8&42 9&51 9&48 7&49 1&60 1&31 6&59 7\\
&R-50&41 14M&63 26G&4 7&54 8&75 9&63 1&53 0&56 2&53 8&55 5&62 4&38 7&62 5\\
&R-101&60 13M&82 74G&3 7&53 8&75 4&61 6&39 0&55 2&52 8&54 3&62 0&38 5&60 4\\
\hline

\multirow{3}{*}{Cascade R-CNN \cite{Cai_2019}}
&R-18&55 93M&77 54G&3 4&52 7&73 4&60 3&\bf 49 0&54 7&50 9&51 4&62 3&34 9&62 3\\
&R-50&68 94M&91 06G&3 0&55 6&75 5&63 8&44 9&57 4&54 4&56 8&63 6&38 7&63 5\\
&R-101&87 93M&110 53G&2 6&56 0&76 1&63 6&51 2&57 5&54 7&56 2&63 9&41 3&62 6\\
\hline

\multirow{3}{*}{Grid R-CNN \cite{lu2019grid}}
&R-18&51 24M&163 15G&3 9&51 9&72 1&59 2&40 4&54 2&50 1&50 7&61 8&33 3&61 9\\
&R-50&64 24M&176 67G&3 4&55 9&75 8&64 3&40 9&57 5&54 8&56 7&62 9&39 5&64 4\\
&R-101&83 24M&196 14G&2 8&55 6&75 6&62 9&45 6&57 1&54 5&55 5&62 9&41 0&62 9\\
\hline

\multirow{3}{*}{RepPoints \cite{yang2019reppoints}}
&R-18&20 11M&\bf 35 60G&5 6&51 7&76 9&57 8&43 8&54 0&49 7&50 8&63 3&33 6&59 2\\
&R-50&36 60M&48 54G&4 8&56 0&80 2&63 1&40 8&58 5&53 7&56 7&65 7&39 3&62 3\\
&R-101&55 60M&68 02G&3 8&55 4&79 0&62 6&42 2&57 3&53 9&56 0&65 8&39 0&60 9\\
\hline 
\hline 
\emph{one-stage:} &&&&&&&&&&&&&& \\
\multirow{3}{*}{RetinaNet \cite{Lin2017Focal}}
&R-18&19 68M&39 68G&7 1&44 7&66 3&50 7&29 3&47 6&42 5&46 9&54 2&23 9&53 8\\
&R-50&36 17M&52 62G&5 9&49 3&70 3&55 4&36 5&51 9&47 6&54 4&56 6&27 8&58 3\\
&R-101&55 16M&72 10G&4 5&50 4&71 7&57 3&34 6&52 8&49 0&54 6&57 0&33 7&56 3\\
\hline 

\multirow{3}{*}{FreeAnchor \cite{2019arXiv190902466Z}}
&R-18&19 68M&39 68G&6 8&49 0&71 9&55 3&38 6&51 7&46 7&47 2&62 8&28 6&57 6\\
&R-50&36 17M&52 62G&5 8&54 4&76 6&62 5&38 1&55 7&53 4&55 3&65 2&35 3&61 8\\
&R-101&55 16M&72 10G&4 4&54 6&76 9&62 9&36 5&56 5&52 9&54 0&65 1&38 4&60 7\\
\hline 

\multirow{3}{*}{FoveaBox \cite{DBLP:journals/corr/abs-1904-03797}}
&R-18&21 20M&44 75G&6 7&51 6&74 9&57 4&40 0&53 6&49 8&51 0&61 9&34 6&59 1\\
&R-50&37 69M&57 69G&5 5&55 3&77 8&62 3&44 7&57 4&53 4&57 9&64 2&36 4&62 8\\
&R-101&56 68M&77 16G&4 2&54 7&77 3&62 3&37 7&57 1&52 4&55 3&63 6&38 9&60 8\\
\hline 

\multirow{3}{*}{PAA \cite{2020arXiv200708103K}}
&R-18&\bf 18 94M&38 84G&3 0&52 6&75 3&58 8&41 3&55 1&50 2&49 9&64 6&35 6&60 5\\
&R-50&31 89M&51 55G&2 9&56 8&79 0&63 8&38 9&58 9&54 9&56 5&66 9&39 9&64 0\\
&R-101&50 89M&71 03G&2 4&56 5&78 5&63 7&40 9&58 7&54 5&55 8&66 5&42 0&61 6\\
\hline 

\multirow{3}{*}{FSAF \cite{zhu2019feature}}
&R-18&19 53M&38 88G&\bf 7 4&49 6&74 3&55 1&43 4&51 8&47 5&45 5&63 5&30 3&58 9\\
&R-50&36 02M&51 82G&6 0&54 9&79 3&62 1&46 2&56 7&53 3&53 7&66 4&36 8&62 5\\
&R-101&55 01M&55 01G&4 5&54 6&78 7&61 9&46 0&57 1&52 2&53 0&66 3&38 2&61 1\\
\hline 

\multirow{3}{*}{FCOS \cite{DBLP:journals/corr/abs-1904-01355}}
&R-18&\bf 18 94M&38 84G&6 5&48 4&72 8&53 7&30 7&50 9&46 3&46 5&61 5&29 1&56 6\\
&R-50&31 84M&50 34G&5 4&53 0&77 1&59 9&39 7&55 6&50 5&52 3&64 5&35 2&60 0\\
&R-101&50 78M&69 81G&4 2&53 2&77 3&60 1&43 4&55 4&51 2&51 7&64 1&38 5&58 5\\
\hline 

\multirow{3}{*}{ATSS \cite{zhang2019bridging}}
&R-18&\bf 18 94M&38 84G&6 0&54 0&76 5&60 9&44 1&56 6&51 4&52 6&65 5&35 8&61 9\\
&R-50&31 89M&51 55G&5 2&58 2&\bf 80 1&66 5&43 9&60 6&55 9&\bf 58 6&67 6&41 8&64 6\\
&R-101&50 89M&71 03G&3 8&57 6&79 4&65 3&46 5&60 3&55 0&57 7&67 2&42 6&62 9\\
\hline 

\multirow{3}{*}{GFL \cite{li2020generalized}}
&R-18&19 09M&39 63G&6 3&54 4&75 5&61 9&35 0&57 1&51 8&51 8&66 9&36 5&62 5\\
&R-50&32 04M&52 35G&5 5&\bf 58 6&79 3&\bf 66 7&46 5&\bf 61 6&55 6&\bf 58 6&\bf 69 1&41 3&\bf 65 3\\
&R-101&51 03M&71 82G&4 1&58 3&79 3&65 5&45 1&60 5&\bf 56 3&57 0&\bf 69 1&\bf 43 0&64 0\\


\hline 
\end{tabular}
\end{center}
\end{table*}
Therefore, in terms of accuracy, the accuracy difference between the multi- and the one- stage methods in AP is not obvious, and the AP$_{S}$ of different methods is always the lowest among the three size AP  For class AP, AP$_{Sc}$ lags significantly behind the other three classes because it has the smallest number of instances  In terms of efficiency, large parameters and FLOPs result in low FPS on AGX, with a maximum FPS of 7 4, which is hardly deployable on underwater robot  Finally, we also found that ResNet101 was not significantly improved over ResNet50, which means that a very deep network may not be useful for detecting small creatures in underwater scenarios  

Consequently, the design of high accuracy and high efficiency detector is still the main direction in this field and there is still large space to improve the performance 
In order to achieve this goal, a shallow backbone with strong multi-scale feature fusion ability can be proposed to extract the discriminant features of small scale aquatic organisms; a specially designed training strategy may overcome the DUO's long-tail distribution, such as a more reasonable positive/negative label sampling mechanism or a class-balanced image allocation strategy within a training batch 

\section{Conclusion}
In this paper, we introduce a dataset (DUO) and a corresponding benchmark to fill in the gaps in the community  DUO contains a variety of underwater scenes and more reasonable annotations  Benchmark includes efficiency and accuracy indicators to conduct a comprehensive evaluation of the \emph{SOTA} decoders  The two contributions could serve as a reference for academic research and industrial applications, as well as promote community development 
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ground_truth: URPC2017, URPC2018, URPC2019, URPC2020_ZJ and URPC2020_DL.